' 



-M 



f^I 



' 1 . 



( 



KEY 



TO A 



TREATISE ON SURVEYING 



BY 



/ 



SAMUEL ALSOR 

AUTHOR OP "A TREATISE ON ALGEBRA," ETC. 




PHILADELPHIA: 
E. C. & J. BIDDLE, No. 508 MINOR STREET, 

(Between Market and Chestnut Sts., west of Fifth St.) 

1857. 



Entered according to Act of Cot,grcsH, in the year 1857, by 

E. C. & J. BIDDLE, 

in the Clerk's OfBce of the District Court of the United States for the Eastern District of 

Pennsylvania. 



STEREOTYPED BY L. JOHNSON & CO. 
PHILADELPHIA. 




LC Control Nuinber 





tmp96 025490 



KEY 



TO 



ALSOP^S SURVEYING. 





CHAPTEE 


III. 




PLANE TRIGONOMETRY. 




Article 137. 




Ex. 3. Here, 


(Eule 3,) 






As AB 


59.47 




A. C. 8.225702 


: BC 


48.52 




1.685921 


; ; rad. 






10.000000 


: tan. A 


39° 12' 36^' 




9.911623 


And (Rule 2) 


' 






As COS. A 


39° 12' 36''. 




A. C. 0.110791 


: rad. 






10.000000 


:: AB 






1.774298 


: AC 


76.75 




1.885089 


Ex. 4. Here, 


(Rule 1,) 






As AC 


97.23 




A.C. 8.012200 


: BC 


75.87 




1.880070 


: : rad. 






10.000000 


: sin. A 


51° 17' 22" 




9.892270 

3 



p. 66.] KEY TO ALSOP'S SURVEYING. [Chap. III. 



And (Rule 2) 






As rad. 




A. C. 0.000000 


: COS. A 


51° 17' 22" 


9.796148 


:: AC 




1.987800 


: i\B 


60.81 


1.783948 


Ex. 5. Here, (Rule 3,) 




As tan. A 


42° 19' 24" 


A. C. 0.040636 


: rad. 




10.000000 


: : BC 


25.54 


1.407221 


: AB 


28.045 


1.447857 


And (Rule 1) 






As sin. A 




A. C. 0.171783 


: rad. 




10.000000 


:: BC 




1.407221 


: AC 


37.932 


1.579004 


Ex. 6. Here A 


= 90° - C = 17° 


17' 51" 


Rulel. 






As rad. 




A. C. 0.000000 


: sin. A 


17° 17' 51" 


9.473243 


:: AC 


495 


2.694605 


: BC 


147.18 


2.167848 


And (Rule 2) 






As rad. 




A.C. 0.000000 


: COS. A 




9.979900 


:: AC 




2.694605 


: AB 


472.612 


2.674505 


Ex. 7. Rule 3. 


" 




As rad. 




A. C. 0.000000 


: tan. A 


42° 8' 45" 


9.956659 


:: AB 


63.2 


1.800717 


: BC 

4 


57.20 


1.757376 



Chap. III.] 



PLANE TRIGONOMETRY. 



[Pp. 66-69. 



And (Rule 2) 
As COS. A 
: rad. 
:: AB 

: AC 



85.24 



A. C. 0.129925 
10.000000 

1.800717 
1.930642 



Article 138. 



Ex. 2. AC = x/AB^ + AC^ = x/36 + 64 = ^/100 = 10. 



Ex. 3. BC = v/(AC + AB) . (AC - AB) = ^/77.37 x 18.47 
= x/1629.0239 = 37.8. 

Ex. 4. 49.27 + 37.42 = 86.69 
49.27 - 37.42 = 11.85 

Ans. 32.05 



log. 1.937969 
" 1.073718 

2 )3.011687 
1.505843 



Article 139. 



Ex. 3. Here, (Rule 1,) 
AsAB 
: AC 

: : sin. C 

: sin. B 
A 

And (Rule 2) - 
As sin. C 
: sin. A 
:: AB 
: BC 32.774 



37.25 
42.59 

57° 29' 15'' 
74° 36' 53" 

47° 53' 52" 



A. C. 8.428874 
1.629308 
9.925969 

9.984151 



A.C. 0.074031 
9.870375 
1 .571126 

1.515532 



Ex. 4. Here C = 180 - A + B == 125° 57' 14" 



And (Rule 2) 
As sin. C 
: sin. B 
:: AB 

: AC 



125° 57' 14" 
24° 15' 17" 
325 

164.93 



A. C. 0.091788 
9.613624 

2.511883 
2.217295 



p. 69.] 



KEY TO ALSOP'S SURVEYING. 



[Chap. III. 



Eule 2. 
As sin. C 
: sin. A 
:: J\B 


29° 47' 29" 


A.C, 


. 0.091788 
9.696219 

2.511883 


: BC 


199.48 




2.299890 


Ex. 5. 
As AC 
: .AB 

: : sin. B 


106.49 
127.54 

52° 27' 18'' 


A.C. 


7.972691 
2.105646 
9.899205 


: sin. C 


108° 16' 3" 




9.977542 


As sin. B 
: sin. A 
:: AC 


19° 16' 39" 


A.C. 


0.100795 
9.518703 
2.027309 


: BC 


44.34 




2.646807 


Ex. 6. Eule 1. 

As AC 
: AB 
: : sin. B 


398.47 ' 
527.63 
43° 29' 11" 


A.C. 


. 7.399605 
2.722330 
9.837703 


: sin. C 
or 


65° 40' 44" 
114° 19' 16" 




9.959638 


Eule 2. 
As sin. B 

: sin. A 

:: AC 


C acute. 

43° 29' 11" 

70° 50' 5" 


A.C. 


0.162297 
9.975237 
2.600395 


: BC 


546.93 

C obtuse. , 




2.737929 


As sin. B 
: sin. A 
: : AC 


22° 11' 33" 


A.C. 


0.162297 
9.577169 
2.600395 


: BC 

G 


218.71 




2.339861 



Chap. III.] PLANE TRIGONOMETRY. [P. 71. 



" 


Article 140. 




Ex. 1. 






As AB + BC 


1875.10 


A.C. 6.726976 


: AB-BC 


176.04 


2.245611 


C + A 
: : tan. — - — 

2 


57° 8' 20'' 
8° 16' 10" 


10.189790 


C-A 

: tan. 

2 


9.162377 


C 


= 65'' 24' 30" 




A 


= 4«° 52' 10" 




Hule 2, Art. 139. 






As sin. A 




A. C. 0.123083 


: sin. B 


65° 43' 20" 


9.959787 


:: BC 


849.53 


2.929179 


: AC 


1028.13 


3.012049 


Ex. 3. 






As AB + AC 


373.39 


A. C. 7.427838 


: AB~AC 


61.47 


1.788663 


C + B 
: : tan. — - — 
A 


G^"" 20' 30" 
19° 43' 42" 


10.338123 


C-B 

: tan. — 

A 


9.554624 



C = 85° 4' 12" 

B = 45° 36' 48" 

As sin. C 85° 4' 12" A. C. 0.001610 

: sin. A 49° 19' 9.879855 

: : AB 217.43 2.337320 

: BC 165.495 2.218785 



p. 73.] KEY TO ALSOP'S SURVEYING. [Chap. Ill 





Article 141. 




Ex. 2. 






As AC 




A. C. 6.270585 


: BC 




3.964217 


: : rad. 




10.000000 


; tan. X 


59° 47' 5" 

45° 


10.234802 


tan. X — 45° 


14° 47' 5'' 


9.421483 


A + B 

tan. 2 


58° 21' 18" 
23° 10' 58" 


10.210207 


A-B 
tan. ^ 


9.631690 


A 


81° 32' 16" 




As sin. A 




A. C. 0.004754 


: sin C. 


63° 17' 24" 


9.950993 


:: BC 




3.964217 


: i\B 


8317 


3.919964 


Ex. 3. 






As AB 




A. C. 8.036575 


: BC 




2.416347 


: : rad. 




10.000000 


: tan. x 


70° 35' 9" 
45° 


10.452922 


tan. X — 45° 


25° 35' 9" 


9.680169 


A+ C 
tan. 2 


25° 9' 
12° 40' 11" 


9.671634 


A- C 

tan. — - — 

2 


9.351803 


A 


37° 49' 11" 




As sin. A 




A. C. 0.212413 


: sin. B 


129° 42' 


9.886152 


:: BC 




2.416341 


: AC 


327.27 


2.514912 


8 







ruAP. III.] 


PLANE TRIGOXOME] 

Article 142. 


CRY. [: 


Ex. 2. Here 






As AB 


214 


A.C. 7.669586 


: AC + BC 


362 


2.558709 


: : AC - BC 


28 


1.447158 


: ^T)-DB 


47.364 


1.675453 


Whence AD = 1 


L30.682 and BD = 


83.318. 


Then (Art. 137, 


Rule 2,) 




As AC 


195 


A.C. 7.709965 


: AD 


130.682 


2.116216 


: : rad. 




10.000000 


: COS. A 


47° 55' 13'^ 


9.826181 


And 






AsBC 


167 


A. C. 7.777284 


: BD 


83.318 


1.920739 


: : rad. 




10.000000 


: CQS. B 


60° 4' W 


9.698023 


Ex. 3. 






As AB 


51.67 


A. C. 8.286762 


: AC + CB 


71.11 


1.851931 


: : AC - CB 


16.79 


1.225051 


: AD-DB 


23.107 


1.363744 


Hence AD = 37.3885 and BD = 14.2815. 


Then 






As AC 


43.95 


A. C. 8.357041 


: AD 


37.3885 


1.572738 


: : rad. 




10.000000 


: cos. A 


31° 42' 42'' 


9.929779 


And 






AsBC 


27.16 


A. C. 8.566070 


: BD 


14.2815 


1.154774 


: : rad. 




10.000000 


: cos. B 


58° 16' 34" 


9.720844 

9 



[P. 94. 



r. 76.] 



KEY TO ALSOP'S SURVEYING. 



[Chap. Ill 





Article 143. 




Ex. 2. Here the half-sum of the sides is 288. "Whe 


^^{Ic 


167 


A.C. 7.777284 


214 


" '' 7.669586 




288 


2.459392 


93 


1.968483 


: : rad.^ 




20.000000 


■ : C0S.2 1A 




2)19.874745 


COS. J A 


30° 2' 11" 


9.937372 


A 


60° 4' 22" 




Ex. 3. 






BC 2T.16 






AB 51.6T 




A.C. 8.286762 


AC 43.95 




8.357041 


2)122.78 






J .9 61.39 




1.788098 


J-s-BC 34.23 




1.534407 

2)19.966308 


COS. 1 A 
A 


15° 51' 20" 

31° 42' 40" 


9.983154 



MISCEIiliAMEOUS PROBLEMS. 



Problem 6. — Construction. Draw Fig. i. 

BC (Fig. 1) to represent the tower. 
At B make CBA = 98° 19'. Make 
BB = 5 feet. With B as a centre, and 
a radius of 75, describe an arc cutting 
BA in A. Draw AE and BE parallel 
to BD and AD ; then will AE be the position of the 
instrument. Draw AC, making DAC = 41° 18' 45", and 
C wdll^be the top of the tower. 

10 




Chap, m.] PLANE TRIGONOMETRY. [P. 114. 

Calculation. 

As sin. C 40° 22' 15'' A. C. 0.188605 

: sin. BAG 41° 18' 45" 9.819653 

: : AB 75 1.875061 

: BC 76.44 1.873319 

In this solution, AB is supposed to be equal to BE. 
Without this supposition the calculation would be as 
follows : — 

As EB (75) : EA (5) : : sin. EAB (98° 19') : sin. ABE 
(3° 46' m'^) ; 

And As sin. EAB : sin. AEB (77° 54' 4") : : EB : AB 
(74.11); 

Then As sin. C (40° 22' 15") : sin. BAG (41° 18' 45") 
:: AB :BG = 75.54. 



Fig. 2 

Problem 7. Fig. 2. 
Here AGD = BDG - BAG = 
14° 27' 15". Then, 




As sin. AGD 


14° 27' 


15" 


A.G 


. 0.602745 


; sin. DAG 


24° 18' 






9.614385 


: : AD 


100 






2.000000 


: DG 








2.217130 


COS. GDB 


38° 45' 


15" 




9.892005 


GB 


128.57 






2.109135 


DG 








2.217130 


sin. GDB 








9.796560 


BG 


103.2 






2.013690 


Height of inst. 


4.75 









107.95 = height required. 



n 



p. 114.] 



KEY TO ALSOP'S SURVEYING. 



Problem 8. 

First. To find AD. 

As sin. CAD 15° 51' 45'' A. C. 0.563314 

: sin. ACD 103° 47' 9.987310 

: : CD . 35.75 1.552276 \ 

: AD 2.103900 



Second. To find ED. 
As sin. CBD 24° 7' 
: sin. BCD 45° 29' 30" 
: : CD 35.75 

: BD 



A. C. 0.388706 
9.853180 
1.553276 

1.795162 




Third. To find ABD and AB. 

As BD A. C. 8.204838 

: AD 2.103900 

: : rad. 10.000000 

: tan. x 63° 50' 22" 10.308738 



And 
As rad. A. C. 0.000000 

: tan. x - 45° 18° 50' 22" 9.533005 

ABD + BAD 



: : tan. 

: tan. 
ABD 



ABJ) -BAD 



64° 58' 52J" 



36° 10' lOJ" 



101° 9' 3" 



10.330956 



9.863961 



As sin. ABD 
: sin. ADB 

:: AD 
: AB 



A.C. 0.008277 

50° 2' 15" 9.884492 

2.103900 

99,236 1.996669 



12 



Chap. IH.] 



PLANE TRIGONOMETRY. 



Problem 9. Fig. 75. Surveying. 
First. To find AD. 

As sin. CAD 
: sin. ACD 
:: CD 
: AD 



61° 18' 
68° 15' 

7.75 



[P. 114. 



A. C. 0.056928 
9.967927 
0.8893Q2 

0.914157 



Second. To find BD and BAD. 



As sin. DBE 


61° 20' A. C. 0.056790 


: sin. DEB 


.75° 10' 9.985280 


:: DE 


7.92 0.898725 


: BD 


0.940795 


J\D 


0.914157 


tan. X 


46° 45' 22" 10.026638 


As rad. 


A. C. 0.000000 


: tan. x — 45° 


1° 45' 22" 8.486565 


BAD + ABD 

: : tan. 


33° 37' 9.822703 


BAD - ABD 

: tan. 

2 


1° 10' 4" 8.309268 


BAD 


34° 47' 4" 


Lastly. To find AB. 




As sin. BAD 


34° 47' 4" A. C. 0.243752 


: sin. BDA 


112° 46' 9.964773 


:: BD 


0.940795 


: ^B 


14.10 ch. 1.149320 



13 



p. 116.] 



KEY TO ALSOP'S SURVEYING. 



Problem 10. The construction is as 
in Problem 4, page 112, Surveying. 

First. To find BAG. 
EC 4153 

AC 5916 A. C. 6.227972 

AB 4596 , " " 6.337620 



2)14665 



7332.5 
3179.5 



COS. 1 BAG 22° 11' V 
BAG 44° 22' 2'' 



3.865252 

3.502359 

2)19.933203 

9.966601 




-'-^D 



Second. To find AE and ABE. 



As sin. AEG 


136° 26' 30" 


A. G. 0.161722 


: sin. AGE 


19° 14' 30" 




9.517926 


:: AG 






3.772028 


: AE 






3.451676 


AB 


4596 




3.662380 


tan. X 


58° 23' 1" 




10.210704 


As rad. 


A.G. 


0.0000000 


: tan. x — 45° 


13° 23' 1" 




9.376451 


AEB + 

: : tan. 

2 


ABE 

55° 39' 29" 

ABE 

19° 12' 2" 




19.165435 


AEB 
: tan. 

2 


9.541886 



ABE 36° 27' 27" 

BAD = ABE - ADB == 17° 12' 57". 



Third. To find AD and BD. ' 
As sin. ADB 19° 14' 30" A. G. 0.482074 

: sin. ABD 143° 32' 33" 9.773952 

:: AE 3.662380 

: AD 8287.2 3.918406 

u 



Chap. III.] 



PLANE TRIGONOMETRY. 



[P. 115, 



And 








As sin... ABD 




A.C. 


0.482074 


: sin. B^D 


1T° 12' 5T'' 




9.471251 


:: AB 






3.662380 


: BD 


4127.7 




3.615705 


Lastly. To find DC. 






As sin. ADC 


43° 33' 30'' 


A.C. 


0.161723 


: sin. CAD 


61° 34' 59" 




9.944240 


:: AC 


5916 




3.772028 


: CD 


7550.8 




3.877991 



Fig. 5. 



Problem U. 

First. To find CA and AG. 










B 


2::^^ 


As sin. CAD 


21° 48' 45" 




A.C. 


0.429959 


: sin. ADC 


49° 48' 30" 






9.883030 


:: CD 


800 






2.903090 


: CA 








3.216079 


sin. ACQ 


47° 22' 15" 






9.866732 


AG 


1210.07 






3.082811 



Second. To find BC and BE. 



As sin. CBD 


5° 45' 




A.C. 


0.999184 


: sin. BDC 


6° 






9.019235 


:: CD 


800 






2.903090 


: CB 








2.921509 


sin, ECB 


12° 30' 






9.335337 


EB 


180.65 






2.256846 


AG 


1210.07 








AF 


1390.72 -= 


height 


required. 



15 



IM15,] KEY TO ALSOP'S SURVEYING. [Chap. III. 

Problem 12. Fig. 76. Surveying. — The measured angles 
being the horizontal and vertical angles, we have the follow- 
ing data, — AJ3EF being a horizontal plane : — AB = 252.28 
feet, AEE = 80° 51' 30'^ BAE = 82° 54' 30'', ABF = 
74° 37', BAF = 89° 24', and the elevations as in the 
problem. 

First. To find AE, AF, AFE, and EF. 

As sin. AEB 16° 14' A. C. 0.553541 

: sin. ABE 80° 51' 30" 9.994448 

: : AB 252.28 2.401883 



: AE 






2.949872 


And 








As sin. AFB 


15° 59' 


A.C 


. 0.560103 


: sin. j^BF 


74° 37' 




9.984155 


:: AB 






2.401883 


: AF 






2.946141 


AE 






2.949872 


tan. X 


45° 14' 46" 




10.003731 


tan. X — 45° 


0° 14' 46" 




7.633005 


AFE + AEF 
tan. 2 


86° 45' 15" 
4° 19' 53" 




11.246335 


AFE- ^EF 

tan. ^ 


8.879340 


Then 






As sin. AFE 


91° 5' 8" 


A.C 


. 0.000078 


: sin. EAF 


6° 29' 30" 




9.053304 


:: ^E 






2.949872 


: EF 


100.75 




2.003254 



Second. To find EC, EH, and CH. 

As rad. A.C. 0.000000 

: tan. EAC 3° 45' 8.816529 

: : EA 2.949872 

: EC 58.40 1.766401 



AP. III.] 




PLANE TRIGONOMETRY. [Pp. 115, 116, 


And 






As rad. 




A. C. 0.000000 


: tan. 


EAH 


9° 25' 9.219710 


:: EA 




2.9498T2 


: EH 




14T.7T 2.169582 


EC 




58.40 


CH 




89.37 



Third. To find FD, EG, and DG. 
As rad. 
: tan. DAE 3° 54' 
:: EA 
: YD 60.22 



A.C. 0.000000 

8.833613 
2.946141 

1.779754 



And 
As rad. 
: tan. GAE 
:: EA 

: EG 
ED 

DG 



10° 29' 30" 

163.59 

60.22 



103.37 



A. C. 0.000000 
9.267614 
2.946141 



CD is sensibly equal to EE = 100.75. 



2.213755 



Problem 13. Eig. 77. Surveying. 



First. To find AE and EB. 






As sin. AEB 


42° 40' 


A.C. 


0.168942 


: sin. ABE 


72° 43'- 




9.979934 


:: AB 


7.37 




0.867467 


: AE 


10.38 




1.016343 


And 








As sin. AEB 




A.C. 


0.168942 


: sin. BAE 


64° 37' 




9.955909 


:: AB 






0.867467 


: BE 


9.8247 




0.992318 



17 



IM16.] KEY TO ALSOP'S SURVEYING. [Chap. TIT. 



Second. To find CE. 








As BE + BC 




1T.774T 


A.C. 


8.726427 


: BE-BC 




.8747 




-1.941859 


BCE + 
: : tan. ^ 


BEC 
BEC 


15° 14' 
43' 37'' 




9.435078 


BCE- 
• toil 


8 1 08.S6-1- 


. tan. ^ 




CJ. i.\JtJfJ\J^ 


BCE 


15° 57' 37" 




BEC 




14° 30' 23" 






And 










As sin. BEC 




14° 30' 23" 


A.C. 


0.601213 


: sin. CBE 




149° 32' 




9.705040 


:: BC 




8.95 




0.951823 


: CE 








1.258076 



Third. To find DF and CF. 



As sin. DEC 


27° 47' 


A.C. 


0.331494 


: sin. DCF 


69° 38' 




9.971964 


: : DC 


9.33 




0.969882 


: DF 


18.765 




1.273340 


And 








As sin. DEC 




A.C. 


0.331494 


: sin. CDF 


82° 35' 




9.996351 


:: DC 


- 




0.969882 


: CF 






1.297727 



18 



:hap. Ill] 



PLANE TRIGONOMETRY. 



[P. 1]<1. 



Fom-tli. To find EF. 
CF 
CE 



1.29772T 

1.258076 



tan. X 




47° 36' 43'' 

45° 


10.039651 


tan. X — 45° 




2° 36' 43" 


8.659140 


CEF + CFE 


28° 5' 19" 
1° 23' 41" 


9.727293 


tan. ^ 


CFE 


CEF- 

fin 


8.386433 


2 




CEF 


29° 29' 0" 




CFE 




26° 41' 38" 




As sin. CEF 




29° 29' 


A. C. 0.307885 


: sin. ECF 




123° 49' 23" 


9.919476 


:: CF 






1.297727 



: EF 33.50 1.525088 

AEF = AEB + BEC + CEF = 86° 39' 23". 
EFD = CFD + CFE = 54° 28' 38". 



Problem 14. 

First. To find AH and BH. 
As rad. A. C. 0.000000 

: COS. BAH 3° 14' 30" 9.999304 
: : AB 850 2.929419 

: AH = Fa 2.928723 



Fig. 6. 



E 




And 
As rad. 
: sin. BAH 
:: AB 

: BH 



A. C. 0.000000 
8.752412 
2.929419 



48.07 



1.681831 

19 



p. 116.] 



KEY TO ALSOP'S SURVEYING. 



[Chap. III.' 



Second. To find GE and FE. 



As sin. GEF 


21° 27' 


30" 


A.C. 


0.436727 


: sin. EFG 


87° 49' 






9.999685 


:: FG 








2.928723 


: GE 


2318.1 






3.365135 


And 










As sin. GEF 






A.C. 


0.436727 


. : sin. FGE 


70° 43' 


30" 




9.974947 


:: FG 








2.928723 


: FE 


2189.8 






3.340397 



Third. To find FD, GD, AF, and BG. 



As sin. FDG 


44° 32' 


A. C. 0.154081 


: sin. FGD 


47° 39' 


9.868670 


:: FG 




2.928723 


: FD 


894.3 


2.951474 


And 






As sin. FDG 




A. C. 0.154081 


: sin. DFG 


87° 49' 


9.999685 


:: FG 




2.928723 


: DG 


1209.2 


3.082489 


DF 




2.951474 


tan. A DP 


3° 25' 45" 


8.777584 


AF 


53.59 


1.729058 


BH 


48.07 




BG 


101.66 




Fourth. To find CK and CE. 




AK 




3.340397 


tan. CAK 


35° 27' 


9.852466 


CK 


1559.1 


3.192863 


AF 


53.59 




CE 


1612.69 




20 







CHAPTER IV. 



CHAIN SURVEYING. 



Ex. 2. 
17.25 + 16.43 + 14.65 



Article 251. 



48.33 



2 



j- sum 
j- sum 



sum 
sum 



-17.25 
-16.43 
-14.65 



2 
24.165 
6.915 

7.735 
9.515 



= 24.165 = half sum of sides. 



1.383187 
0.839792 
0.888460 
0.978409 

2 )^4.089848 
Area, 110.9 eh. = 11 A., E., 14.4 P. 2.044924 



Ex. 3. 



19.58 + 16.92 + 12.76 



-19.58 
-16.92 
-12.76 



2 



24.63 
5.05 
7.71 

11.87 



49.26 

-— - = 24.63 = 1 5. 

2 ^ 

1.391464 
0.703291 
0.887054 
1.074451 



106.69 ch. = 10 A., 2 R., 27.04 P. 



2)4.056260 
2.028130 



Article 254. ' 

Ex. 2. Fig. 107. Surveying. 
2AED =AE.ED = 4.15 x 8.25 = 34.2375 

2EFCD = EF.(ED + CF) = 5.02 x 17.61 = 88.4022 
2 CFB = FB.FC = 2.26 x 9.36 = 21.1536 



2) 143.7933 

71.89665 ch. 



7 A. 



R., 30.3 P. 



21 



Pp. 146, 140.] KEY TO ALSOP'S SURVEYING. 



[Chap. IV. 



Fig. 7 




Ex. 3. 



A E B F 

2AEC =AE.EC = 3.76 x 5.68= 21.3568 

2 CEFD = EF . (CE + FD) = 9.89 x 14.60 = 144.3940 

165.7508 
2BDF =BF.FD = 3.98 x 8.92= 35.5016 



2) 130.2492 
65.1246 ch. 



= 6 A., 2 R., 1.99 P. 

Article 265. 



Fio-. 8. 




First field. 





Bases. 


Perpen- 
diculars. 


Areas. 

109.8900 
102.4320 


2ABCF 
2FCDE 


' 9.90 
9.70 


11.10 
10.56 



2 )212.3220 
10 A., 2 R., 18.576 P. = 106.161 ch. 



Second field. 



2EKGD 


13.15 


8.75 


115.0625 


2e:ghi 


12.85 


7.20 


92.5200 


2GIHL 


11.50 


4.32 


49.6800 



2) 257.2625 
12 A., 3 B., 18.1 P. = 128.63125 ch. 



22 



Chap. IV.} 



CHAIN SURVEYING. 



[Pp. 149,151. 



Fig. 9. 




First field. 



Areas. 


Bases. 


Perpen- 
diculai'S. 


Areas. 


2 ABC ' 

2ACDE 
2AEFG 


11.50 
12.55 
10.51 


4.82 

12.65 

9.20 


55.4300 

158.75T5 
96.6920 



2 )310.8795 
15 A., 2 E., 7 P. = 155.43975 ch. 



Second field. 



2FaHL 
2HIKL 


13.95 

10.80 


13.00 
10.85 


181.3500 
117.1800 



2 )298.5300 
14 A., 3 E., 28.24 P. = 149.265 ch. 



Ex. 2. First tract. 



Article 256. 





Dis- 
tances. 


Offsets. 


Inter. 
Dist. 


Sum of 

Offsets. 


Double Areas. 




0.00 


0.00 








Left baud 


7.90 


6.00 


7.90 


6.00 


47.4000 


areas. 


19.20 


4.00 


11.30 


10.00 


113.0000 




21.45 


3.75 


2.25 


7.75 


17.4375 




31.27 


0.00 
0.00 


9.82 


3.75 


36.8250 




0.00 










2.00 


4.65 


2.00 


4.65 


9.3000 


Eisjlit liand 


8.40 


5.16 


6.40 


9.81 


62.7840 


areas. 


14.85 


5.23 


6.45 


10.39 


67.0155 




20.70 


6.42 


5.85 


11.65 


68.1525 




25.90 


4.76 


5.20 


11.18 


58.1360 




31.27 


0.00 


5.37 


4.76 


25.5612 



25 A., 1 E., 4.89 P. 



2 )505.6117 
= 252.80585 ch. 

23 



Pp. 151, 157.] KEY TO ALSOP'S SURVEYING. 

Second tract. 



[Chap. IV. 



Dis- 
tances. 


Offsets. 


Inter. 
Dist. 


Sum of 
Offsets. 


Double Areas. 


0.00 


0.00 








1.75 


2.37 


1.75 


2.37 


4.1475 


10.26 


7.29 


8.51 


9.66 


82.2066 


21.37 


4.35 


11.11 


11.64 


129.3204 


28.46 


5.94 


7.09 


10.29 


72.9561 


40.25 


0.00 


11.79 


5.94 


70.0326 


0.00 


0.00 




4.29 


6.23 


4.29 


6.23 


26.7267 


15.48 


3.19 


11.19 


9.42 


105.4098 


32.54 


8.26 


17.06 


11.45 


195.3370 


36.17 


7.92 


3.63 


16.18 


58.7334 


40.25 


0.00 


4.08 


7.92 


32.3136 



2 )777.1837 
38 A., 3 K., 17.47 P. - 388.59185 ch. 



Article 257. 



Ex. 2. 

First. To find ABC. 



21.69 + 23.72 + 14.50 




AB 
BC 
AC 



ABC 



29.955 

• 8.265 

6.235 

15.455 

154.457 



1.476469 
0.917243 
0.794836 
1.189068 
2)4.377616 
2.188808 



24 



Chap. IV.] CHAIN SURVEYING. [P. 157. 

Second. To find ABE. 

21.69 + 13.96 + 13.44 49.09 ^ , 





2 




2 


— ^d-:r.,tjnLKJ, 


1 9 

2 "^ 




24.545 




1.389963 


Is - AB 




2.855 




0.455606 


J-s - BE 




10.585 




1.024691 


J 5 - ^E 




11.105 




1.045518 
2)3.915778 



ABE 90.759 1.957889 



= 26.22. 

1.418633 
0.397940 
0.945469 
1.173186 

2 )3.935228 

CBD 92.814 1.967614 



25 



To find CBD. 




23.72 + 17.40 + 11.32 


: 52.44 


2 


2 


1 9 

2 "^ 


26.22 


is- CB 


2.50 


is- BD 


8.82 


is - CD 


14.90 



V. I'jT.] 

Offsets. 



KEY TO ALSOP'S SURVEYING, 



[Chap. IV 



Base. 


Distance. 


Offsets. 


Int. Dist. 


Sum of 
Offsets. 


Double 
Areas. 




000 










4.80 


.75 


4.80 


.75 


3.6000 


CD 


7.39 


.97 


2.59 


1.72 


4.4548 




8.96 


.72 


1.57 


1.69 


2.6533 




10.54 


.55 


1.58 


1.27 


2.0066 




11.32 


.00 


.78 


.55 


.4290 




5.51 








6.52 


.50 


1.01 


.50 


.5050 




8.24 


.45 


1.72 


.95 


1.6340 


FB 


10.16 


.87 


1.92 


1.32 


2.5344 




12.37 


.35 


2.21 


1.22 


2.6962 




14.14 


.63 


1.77 


.98 


1.7346 




17.40 




3.26 


.63 


2.0538 




24.3017 

.1 




00 










2.95 


.75 


2.95 


.75 


2.2125 


DF 


4.52 


.75 


1.57 


1.50 


2.3550 




5.51 




.99 


.75 


.7425 



Subtractive 



ABC 
ABE 
CBD 



5.3100 
2)18.9917 
9.4958 
154.457 
90.759 
92.814 



Area, 34 A., 3 R., = 347.5258 eh. 



2G 



CHAPTER V. 



COMPASS SURVEYING. 



Article 326. 

Ex. 1. Fig. 127. Surveying. 
To find BD, AD, BAD, and ABD. 



As sin. DBE 


37° hU 


A. 


C. 0.211468 


: sin. DEB 


120° 45' 




9.934199 


: : DE 


18 




1.255273 


: DB 






1.400940 


As sin. Di\E 


46° 35' 


A. 


C. 0.138839 


: sin. AED 


61° 20' 




9.943210 


: : DE 






1.255273 


: AD 






1.337322 


BD 






1.400940 


tan. X 


49° 10' 54" 
4° 10' 54" 




10.063618 


tan. X — 45° 


8.863999 


BAD + ABD 

tan. ^ 


64° 37' 30" 
8° 45' 47" 




10.323947 


BAD - ABD 

tan. ^ 


9.187946 


BAD 


73° 23' 17" 




ABD 


55° 51' 43" = 


= DGC. 


DGF = 180° - 


55° 51' 43" = 


124 


g/ Yl". 


To find DC, DG, and CG. 






As sin. DCE 


46° 35' 


A. 


C. 0.138839 


: sin. DEC 


26° 50' 




9.654558 


: : DE 






1.255273 


: DC 






1.048670 



27 



p. 188.J 



KEY TO ALSOP'S SURVEYING. 



[Chap. V. 



As sin. DGC 


55° 51' 


43" 


A. C. 0.082134 


: sin. DCG 


38° 53' 


17" 


9.797821 


:: DC 






1.048670 


: DG 


8.48 




0.928625 


As sin. DGC 






A. C. 0.082134 


: sin. CDG 


85° 15' 




9.998506 


:: DC 






1.048670 


: CG 


13.47 




1.129310 



Ex. 2. Fig. 120. Surveying. 
To find BC, CA, CAB, and ABC. 



As sin. CBD 


42° 28' 




A. 


C. 0.170593 


: sin. CDB 


47° 29' 






9.867515 


: : DC 


7.00 






0.845098 


: CB 








0.883206 


As sin. CAD 


47° 18' 




A. 


C. 0.133763 


: sin. ADC 


100° 25' 






9.992783 


: : CD 








0.845098 


: AC 








0.971644 


CB 








0.883206 


tan. X 


50° 47' 
5° 47' 


38" 

38" 




10.088438 


tan. X — 45° 


9.006330 


ABC + BAC 

tan. 

2 


61° 7' 
10° 25' 


20" 




10.258336 


ABC - BAC 

tan. ^ 


9.264666 


ABC 


71° 32' 


20" 




BAC 


50° 41', 


40" 







ACF = BFC - BAC = 16° 53' 20". 
DCF = DCA + ACF = 49° 10' 20". 



28 



Chap. V.] COMPASS SURVEYING. [P. 188. 

To find AF and FC. 



As sin. AFC 
: sin. ACF 

: : AC 
: AF 

And as sin. AFC 
: sin. CAF 
: : AC 
: CF 



67° 35' 

16° 53' 20'' 

2.94 



50° 41' 40" 

7.84 



A. C. 0.034124 
9.463171 
0.971644 



0.468939 

A. C. 0.034124 

9.888617 
0.971644 



0.894385 



Fig. 11. 



Ex. 3. Construction. — Take AB' any 
length. At A and B' make the angles 
B'AE, B'AF, AB'E, and AB'F equal to 
the given angles. Join EF: produce it 
to a, making EG = 10.78. Through G 
draw GD parallel to AE, meeting AF in 
D. Draw DB parallel to FB', and BC 
parallel to B'E. Then will AB he the 
line and CD the known base. 



By Art. 316 we have 

AD : AC : : sin. ABD . sin. ACB : sin. ADB . sin. ABC 
whence 




As AD or 



: AC or 

: : rad. 
: tan. x 
tan. 45° — X 

ACD + ADC 



sin. ABD 80° 47' 
sin. ACB 60° 54' 
sin. ADB 52° 43' 
sin. ABC 37° 23' 



A. C. 0.005643 

'' " 0.058602 

9.900722 

9.783292 

10.000000 



tan. 



tan. 



ACD 



ACD - ADC 



29° 15' 10" 
15° 44' 50" 

72° 23' 30" 

41° 37' 11" 
114° 0'41" 



9.748259 
9.450213 

10.498422 
9.948635 

29 



p. 189.] 



KEY TO ALSOr'S SURVEYING. 



[Chap. V 



As 



And (Art. 316) 

sin. CAD 
^ sin. ADD 
j sin. ACD 
\ sin. ADB 
CD 

AB 



35° 13' 
80° 47' 
114° 0'41" 
52° 43' 

10.78 

13.76 



Ex. 4. To find BD, AD, 
and ABD. 



A. C. 0.239073 

" " 0.005643 
9.960691 
9.900722 
1.032619 

1.138748 

Fig. 12. 




As sin. CBD 


54° 25' 




A. 


C. 0.089765 


: sin. BCD 


33° 18' 






9.739590 


: : CD 


15 






1.176091 


: BD 








1.005446 


As sin. CAD 


36° 2' 




A. 


C. 0.230434 


: sin. ACD 


103° 10' 






9.988430 


: : CD 








1.176091 


• : AD 








1.394955 


BD 


- 






1.005446 


tan. X 


67° 48' 
22° 48' 


45" 
45" 




10.389509 


tan. X — 45° 


9.623888 


ABD + BAD 
tan. ^ 


64° 15' 
41° 6'' 


30" 




10.316805 


ABD - BAD 

tan. ^ 


9.940693 



ABD 105° 21' 30" 

GFD = 360° - (FDB + DBG + BGF) = 72° 21' 30". 
FPD = GFD - DDF = 33° 31' 30". 

80 



Thap. v.] compass surveying. [P. 189. 

To find PD and FD. 

As sin. CPD 35° 40' A. C. 0.234280 

: sin. PCD 105° 30' 9.983911 

: : CD 15 1.176091 

: PD 1.394282 

And as sin. PFD 72° 21' 30" A. C. 0.020921 

: sin. FPD 33° 31' 30" 9.742175 

: : PD 1.394282 

: FD 14.37 1.157378 

Ex. 5. Fig. 118. Surveying. 
To find CA, CB, CAB, and CBA. 



As sin. CAD 


41° 15' 


A. 


C. 0.180887 


: sin. CDA 


43° 45' 




9.839800 


: : CD 


10 




1.000000 


: CA 






1.020687 


As sin. CBD 


55° 1' 


A. C. 0.086547 


: sin. CDB 


87° 39' 




9.999635 


: : CD 






1.000000 


: CB 






1.086182 


CA 






1.020687 


tan. X 


49° 18' 15" 
4° 18' 15" 




10.065495 


tan. X — 45° 


8.876584 


CAB + ABC 

tan. ^ 


61° 10' 

7° 47' 6" 




10.259233 


C^B -- ABC 
tan. 

2 


9.135817 


CAB 


68° 57' 6" 




ABC 


53° 22' 54" 







81 



p. 189.] KEY TO ALSOP'S SURVEYING 

To find CE, and AE, and DCE. 

As sin. AEC 

: sin. CAE 
: : AC 

: CE 5.192 



[Chap. V- 



81° 43' 

29° 19' 54'' 



A. C. 0.004554 

9.690076 
1.020687 
0.715317 



And as sin. AEC 
: sin. ACE 
: : AC 

: AE 



68° 57' 6" 



A. C. 0.004554 
9.970011 

1.020687 



9.89 0.995252 

DCE = DCA + ACE = 95° + 68° 57' 6" = 163° 57' 6". 



Ex. 6. To find DC. 

As sin. DCE 45° 56' A. C. 0.143554 
: sin. DEC 67° 25' 9.965353 ^ 

: : DE 9.25 0.966142 

: DC 11.89 1.075049 ■ 



Fig. 13. 



A;2<r ,. ^ • 






^c 



To find BC. 

As DC + DB 
: DC - DB 



16.14 
7.64 



DBC + DCB 
: : tan. 56° 40' 30" 



DBC - DCB 

: tan. : 



DBC 

And as sin. DCB 
: sin. BDC 
: : DB 
: BC 

Deflection of BC 

32 



A. C. 8.792096 
0.883093 

10.182102 



35° 


45' 


5" 




9.857291 


92° 


25' 


35" 




20° 


55' 


25" 


A.C 


0.447182 


m"" 


39' 






9.962890 


4.25 






0.628389 



10.93 

= 92° 25' 35" 



1.038461 
90° = 2° 25' 35". 



Chap, v.] COMPASS SURVEYING 


[Pp. 188, 194, 


To find BAC and AC. 




AsBC + BA 


18.43 


A. C. 8.734475 


: BC ~ BA 


3.43 


0.535294 


BAC + 
: : tan. ^ 


BCA 

lo 12' 47" 

^^^ 18' 38" 


8.325822 


BAC- 

: tan. ^ 


7.595591 


BAC 


1° 26' 20'' 




Bearing of AB 


K 8T° 


E. 


" AC 


K 88° 26' 


E. 


And 
As sin. BAC 


1° 26' 20" 


A. C. 1.600141 


: sin. ABC 


2° 25' 35" 


8.626705 


: : BC 




1.038461 


: AC 


18.42 


1.265307 


Ex.1. 


Article 333. 




As rad. 




A. C. 0.000000 


: sin. bearing 
: : Dist. 


56° 15' 
37.56 


9.919846 
1.574726 


: Dep. 


31.23 


1.494572 


As rad. 




A. C. 0.000000 


: COS. bearing 
: : Dist. 




9.744739 
1.574726 


: Lat. 


20.87 


1.319465 


Ex. 2. 






As Dist. 


52.95 


A. C. 8.276134 


: Lat. 


36.17 


1.558349 


: : rad. 




10.000000 


: COS. bearing 


46° 55' 


9.834483 



33 



pp. 194, 196.] KEY 


TO ALSOP'S SURVEYING. [Cuap. V. 


As rad. 






A. C. 0.000000 


: sin. bearing 






9.863538 


: : Dist. 






1.723866 


: Dep. 


38.67 




1.587404 


Ex. 3. 








As Lat. 


19.25 




A. C. 8.715569 


: Dep. 


26.45 




1.422426 


; : rad. 






10.000000 


: tan. bear. 


53° 57' 




10.137995 


As COS. bear. 






A. C. 0.230260 


: rad. 






10.000000 


: : Lat. 






1.284431 


: Dist. 


32.71 




1.514691 


Ex. 4. 








As sin. bear. 


33° 30' 




A. C. 0.258111 


: rad. 






10.000000 


: : Dep. 


18.33 




1.263162 


: Dist. 


33.21 




1.521273 


As rad. 


^ 




A. C. 0.000000 


: COS. bear. 






9.921107 


: : Dist. 






1.521273 


: Lat. 


-27.69 




1.442380 




Article 334. 




Ex. 4. 






Ex. 5. 


Dist. Lat. 


Dep. 


Dist. 


Lat. Dep. 


20 8.924 


17.899 , 


20 


n.256 16.532 


7 3.123 


6.264 


7 


3.940 5.786 


.4 .178 


.358 




15.196 22.318 


.09 40 


80 






12.265 


24.601 




34 









Chap. V.] 




COMPASS I 


SURVEYING. 


[Pp. 


196,197. 




Ex.6. 


» 




Ex. 7. 




Dist. 


- Lat. 


Dep. 


Dist. 


Lat. 


Dep. 


100 


9.150 


99.580 


90 


81.733 


37.679 


20 


1.830 


19.916 


7 


6.357 


2.931 


3 


.274 


2.987 


.5 


.454 


.209 


.5 


46 


498 


.06 


54 


25 


.06 


5 


60 




88.598 


40.844 



11.305 123.041 

A slight inspection of tlie rejected figures in the columns 
for latitude in the fourth and sixth examples will show that 
the sec0nd decimal should be increased by one in each 
case. 









Article 335. 






Ex.1. 














Bear. 


Dist. 


Lat. 


Dep. 


Bear. 


Lat. 


Dep. 


761° 


30 


7.003 


29.171 


76f° 


6.876 


29.201 




9 


2.101 


8.751 




2.063 . 


8.760 




9.104 


37.922 


8.939 


37.961 




39 


-.132 


+ .033 

37.955 


diff. 


9.104 


37.922 


76° 42' 


8.972 


.165 


.039 




.7 


.161 


.681 


if of diff. 


.132 


.033 




.06 


14 


58 









9.147 38.694 



Ex. 2. 

Bear. Dist. 

37° 60 
3 


Lat. 

47.918 
2.396 


Dep. 

36.109 

1.805 


Bear. 

37i° 

diff. 
i\ diff 


Lat. 

47.760 
2.388 


Dist. 

36.318 
1.816 




50.314 
-.100 


37.914 
+ .132 


50.148 
50.314 


38.134 
37.914 


37° 9' 63 
.4 
.05 


50.214 

.319 

40 


38.046 

.241 

30 


.166 
.100 


.220 
.132 



50.573 38.317 



35 



Pp. 198, 199.] 


KKY TO 


ALSOP'S SURVEYING. [Chap. V. 


Ex. 3. 




• 




Bear. Dist. 


Lat. 


Dep. Bear. 


Lat. Dep. 


29° 15' 100 


87.250 


48.862 29° 30' 


87.036 49.242 


20 


17.450 


9.772 


17.407 9.848 


3 


2.617 


1.466 


2.611 1.477 




107.317 


60.100 


107.054 60.567 




-.035 


+.062 
60.162 difF. 


107.317 60.100 


29<^ 17' 123 


107.282 


.263 .467 


1 


7 .611 


.343 Adiff. 


.035 .062 


• 


05 44 


24 






107.937 


60.529 






Article 336. 




Ex. 3. 








Bearing 


■ m"" V cosine .55750 


sine .83017 




60 


33.4500 


49.8102 




3 


1.6725 


2.4905 


* 


.4 


.2230 


.3321 




.08 


466 


664 






35.3921 


52.699.2 


Ex. 4. 








Bear. 


52° 49' 


COS. .60437 


sin. .79671 




100 


60.437 


79.671 




30 


18.131 


23.901 




6 


3.626 


4.780 




.7 


.423 


.558 




.05 


.030 


40 






82.647 


108.950 


Ex. 5. 








Bear. 


23° 47' 


COS. .91508 


sin. .40328 




10 


' 9.151 


4.033 




3 


2.745 


1.210 




.6 


.549 


.242 




.02 


.018 


8 






12.463 


5.493 



86 



Chap. V.] 



COMPASS SURVEYING. 



[Pp. 209-211. 



Ex. 2. 



Article 338. 



i Bearings. 


Dist. 
12.93 


N. 


S. 
8.48 


E. 


W. 


Cor. 
S. 


Cor. 
E. 


N. 


S. 


E. 


W. 


; S. 49° W. 




9.76 








8.48 




9.76 


S. 88° W. 


13.68 




.47 





13.67 

e.oT 


1 
1 








.48 




13.67 


N. 2514 W. 


14.09 


12.74 
10.71 


12.73 






6.01 ' 


N.43i^E. 


14.70 


10.07 


1 





10.70 




10.07 




N.r2i^W. 


17.95 
17.68 


17.52 




3.89 


1 


1 


17.51 






3.88 


N.883^E. 


.39 




17.68 




1 


1 


.38 




17.69 




S. 361^ E. 


35.80 




28.78 


21.29 


1 


1 




28.79 


21.30 




s. 7714 W. 


16.15 




3.56 




15.75 


1 


1 




3.57 




15.74 



142.98 41.36 41.29 49.04 49.08 



Ex. 3. 

AB K 39° W. 

75° 26' 



Article 345. 

Ex.4. 

S. 63° 29' E. 
29° 17' 



114° 26' 

180° 



BC S. eS"" 34' w. 



Ex. 5. 

S. 34° 12' E. 

75° 32' 



S. 34° 12' E. S. 41° 20' W. 



Article 346. 




Ex. 5. Ex. 6. 


Ex. 7. 


K 46° 30' E. S. 63° W. 


K 821° w. 


S. 63° 29' W. N.29° W. 


N. 291° w. 


16° 59' - 92° 


531° E. 


180° 180° 




163° I'L. 88° E. 




Article 347. 




Ex. 1. Ex. 2. 


Ex. 3. 


BA S. 87° W. BA ^. 63° W. 


DC S. 15° E. 


BC S. 25° W. BC K 56° E. 


DE K 56° W 


ABC 62° ABC 119° 


41° 




180° 




CDE 139° 



87 



p. 212.] . KEY TO ALSOP'S SURVEYING. [Chap. V. 

Article 348. 

Ex. 2. 

1st. K 43° 25' W. 2d. North. 3d. S. 80° E. 
K 29° 48' E. K 29° 48' E. 

K. 73° 13' W. 109° 48' 

180° 



K 70° 12' E. 

4th. K 89° 55' E. 5th. S. 10° 13' E. 6th. K 63° 55' W. 

N. 29°48'E. K29°48'E. ]^.29°48'E. 



K 60° 7' E. S. 40° 1' E. 93° 43' E. 

180° 



S. 86° 17' W. 



7th. S. 63° 45' W. 8th. K 57° 35' W. 
K 29° 48' E. K 29° 48' E. 





S. 33° 57' W. 


K 87° 23' W, 






Ex. 


3. 








1st. 


S. 63° E. 


2d. S. 47° E. 


3d. 


S. 591° w, 




]^. 841° w. 


N. 841° ^y. 




1^. 841° w, 




S. 211° "W". 


S. 371° W. 


143|° 










180° 




IS^. 361° W. 


5th. 


, K12° W. 


6tkK 17i°E. 


7th, 


. S. 29|-° W. 




K 841° w. 


K 841° w. 




K 841° w. 




K 721° E. 


102° 


1141° 




• 


180° 




180° 




S. 78° E. 


K 65r W. 



38 



Chap. V.] 



COMPASS SURVEYING. 



[P. 216. 



Article 350. 

Ex. 3. 



Sta. 


Bearings. 


Dist. 


N. 


S. 


E. 


w. 


i 1 


S. 29f ° E. 


3.19 




2.77 


1.58 




2 
3 
4 

5 

6 

7 
8 


S. 371° W. 


5.86 
11.29 




4.66 




3.55 


S. 391° E. 




8.74 


7.14 




K53° E. 


19.32 


11.63 




15.43 








(13.92) 




(8.39) 


S. 60}° W. 


7.12 


. 


3.48 




6.21 


S. 29J° E. 


2.18 




1.90 


1.07 




S. 601° w. 


8.12 




4.00 




7.07 



25.55 25.55 25.22 25.22 



As Lat. 13.92 

: Dep. 8.39 
: : rad. 

: tan. bearing K 31° 5' W. 

As COS. bearing 

: rad. 
: : Lat. 

: Dist. 16.26 

Ex.4. 



A. C. 8.856361 

0.923762 

10.000000 

9.780123 

A.C. 0.067315 

10.000000 
1.143639 



1.210954 



Sta. 
1 

2 

o 
O 

4 

5 
6 

7 


Bearings. 


Dist. 


N. 


S. 


E. 


w. 


K46°40'W". 


18.41 


12.63 




10.95 


13.39 


JSr. 54° 30' E 


13.45 


7.81 










(12.61) 




(1.20) 


S. 74°55^E. 


17.58 
15.86 




4.57 


16.97 


S. 47° 50' E. 




10.65 


11.76 




S. 47° 25' W. 


16.36 




11.07 


12.05 


S. 62° 35' W. 


14.69 




6.76 




13.04 



33.05 33.05 39.68 39.68 

39 



p. 216.] 



KEY TO ALSOP'S SURVEYING. 



[ClIAP. V. 



As Lat. 12.61 

: Dep. 1.20 

: : rad. 

: tan. bearing 5° 26' 



A.C. 8.899285 

0.079181 

10.0000 00 

8.978466 



As COS. bearing 

: rad. 
: : Lat. 

: Dist. 



12.67 



A. C. 0.001956 

10.000000 

1.100715 

1.102671 



Ex. 5. 



Sta. 


Bearings. 


Dist. 


N. 


S. 


E. 


w. 


AC 


S. 49J° W. 


9.30 




6.07 




7.05 


CD 
DE 
EB 


S... 321° E. 


10.25 


8.64 


5.51 




S. 5J°W. 


6.75 




6.72 




.62 


K 79f ° E. 


8.10 


1.44 




7.97 




BA 






(19.99) 






(5.81)- 



21.43 21.43 13.48 13.48 



As Lat. 19.99 

: Dep. 5.81 

: : rad. 

: tan. bearing 16° 12' 

As cos. bearing 

: rad. 
: : Lat. 

: Dist. 20.82 



A. C. 8.699187 

0.764176 

10.000000 



9.463363 

A. C. 0.017596 

10.000000 

1.300813 

1.318409 



40 



Chap. V.] 



COMPASS SURVEYING. 
Ex. 6. 



[Pp. 217, 218. 



Sta. 
1 

3 
4 


Beavings. 


Dist. 


N. 


s. 


E. 


w. 


S. 271° E. 


12.65 




11.22 


5.84 




S. 101° E. 


23.45 




23.08 


4.17 




S. 14° W. 


124.33 




120.64 




30.08 


S. 67° E. 


82.43 




32.21 


75.88 
28.17 




5 


S. 17° E. 


96.35 




92.14 










(279.29) 






(83.98) 



279.29 114.06 114. 


As Lat. 279.29 


A. C. 7.553945 


: Dep. 83.98 


1.924176 


: : rad. 


10.000000 


: tan. bearing S. 16° 44' E. 


9.478121 


As COS. bearing 


A. C. 0.018791 


: rad. 


10.000000 


: : Lat. 


2.446055 


: Dist. 291.64 


2.464856 


Article 351. 




Ex.2. 





sta. 

1 
! 2 

;:> 

4 


Bearings. 


Ch. Bear. 


Dist. 
16.55 
11.48 


N. 
12.68 


S. 


E. 


W. 


N. 47° W. 


N. 40° W. 




10.64 


N. 19° 5' W. 


N. 12° 5' W. 


11.23 




2.40 


N. W. 




15.53 


(14.45) 






(5.69) 


N. 2.3° E. 


N. 30° E. 


9.72 
14.00 


8.42 




4.86 
13.87 




5 
6 


N. 75i° E. 


N. 82i° E. 


1.89 






S. 7° E. 


North. 




(48.67) 





18.73 18.73 



41 



p. 219.] 



KEY TO ALSOP'S SURVEYING. 



[Chap. V. 



As Dist. 
: Dep. 
: : rad. 



15.53 
5.69 



sin. ch. bear. K 21° 30' W. 



Bearing, 



S. 



E. 



K 28° 30' W. 



A. C. 8.808829 

0.755112 

10.000000 



9.563941 



As rad. 

: COS. ch. bear. 
: : Dist. 

: Dep. 14.45 



A.C. 0.000000 
9.968678 
1.191171 
1.159849 



Article 352. 

Ex. 2. 



Sta. 
1 
2 
3 
4 
5 


Bearings. 


Ch. Bear. 


Dist. 


N. 


S. 


E. 


W. 


S. 29i° B. 


S. 9i° W. 


3.19 




3.15 




.53 
5.70 


S. 37i° AY. 


S. 761° W. 


5.86 




1.37 




S. 39i° E. 


South. 


(11.28) 




(11.28) 




N. 53° E. 


S. 871° B. 


19.33 




.76 


19.31 


N. 31° 5' W. 


N. 8° 10' E. 


(16.26) 


(16.10) 




(2.31) 


7.01 


6 

7 


S. 60i° w. 


N. 80° W. 


7.12 


1.24 


2.15 


S. 29i° E. 


S. 10° W. 


2.18 
8.12 






.38 j 


8 


S. 60i° W. 


N. 80i° W. 


1.37 






8.00 1 



18.71 



18.71 



21.62 21.62 



As sin. ch. bear. 8° 10' 
: rad. 

: : Dep. 2.31 

: Dist. 16.26 



A. C. 0.847549 

10.000000 

0.363612 



1.211161 



As rad. 

: COS. ch. bear. 
: : Dist. 

: Lat. 16.10 



A.C. 0.000000 
9.995573 
1 .211161 

1.206734 



42 



Chap. V.] 



COMPASS SURVEYING. 



[P. 221. 



Article 353. 



Sta. 
1 

2 
4 
6 

7 
8 


Bearings. 


Dist. 


N. 


s. 


E. 


w. 


S. 29f ° E. 


3.19 




2.77 


1.58 




S. 37J° W. 


5.86 




4.66 


3.55 


F. 53° E. 


19.32 


11.63 




15.43 




S. 60f ° W. 


7.12 
2.18, 




3.48 


1.07 


6.21 


S. 291° E. 




1.90 




S. 601° w. 


8.12 




4.00 




7.07 



11.63 16.81 18.08 
11.63 16.83 

Lat. and Dep. of closing Hue 5.18 1.25 



16.83 



As lat. clos. line 
: Dep. 
: : rad. 

: tan. bear. 



5.18 
1.25 

K 13° 34' W. 



A.C 



. 9.285670 

0.096910 

10.000000 

9.382580 



As COS. bear. 

: rad. 
: : Lat. 

: dist. clos. line 



5.33 



A. C. 0.012290 

10.000000 

0.714330 

0.726620 



Clos. line 



1 ACB 
ACB 



AB 5.33 
BC 16.26 
AC 11.29 

2)32.88 

16.44 
11.11 

4° 4'. 

8° 8'. 



Fisr. 14. 



A. C. 8.788879 
" " 8.947306 

1.215902 
1.045714 



2) 19.997801 
9.998900 




43 



SOP'S SURVEYING. [Chap. V. 

A.C. 9.273380 
1.052694 
9.150686 

;' 9.476760 

Bear, of BC = 17° 26' + 13° 34' = 31° 0' S.E. 5th side. 
Bear, of CA = 31° + 8° 8' = 39° 8' 'N.W. 3d side. 



Pp. 221, 226.] 


KEY TO AI 


And 




As AB 


5.33 


: AC 


11.29 


: : sin. 


ACB 


: sin. 


ABC 17° 2< 



' 


Article 357. 




Ex. 2. Here ABC = 37° 15'. 




As rad. 




A. C. 0.000000 


: sin. ABC 


37° 15' 


9.781966 


TAB 
'' IBC 


17.25 


1.236789 


10.87 


1.036230 


: ABC 


113.497 


2.054985 


ABC 


56.7485 ch. = 5 A. 


, 2 R., 27.98 P. 


Ex. 3. 






As rad. 




A.C. 0.000000 


: sin. A 


126° 47' 


9.903581 


.. JAB 
•• lAC 


23.56 
16.42 


1.372175 
1.215373 


: 2 ABC 


309.834 


2.491129 


ABC 


154.917 ch. = 15 A. 
Article 358. 


, 1 R., 38.67 P. 


Ex. 2. 






As / ''^'^• 
^' I sin. C 




A. C. 0.000000 


42° 21' 


" " 0.171561 


( sin. A 
* tsin. B 


63° 52' 
73° 47' 


9.953166 
9.982367 


TAB 
•' lAB 


17.63 


1.246252 
1.246252 



: 2 ABC 397.7385 2.599598 

ABC 198.8692 ch. = 19 A., 3 R., 21.9 P. 



44 



Chap. V.] 



COMPASS SURVEYING. 



[Pp. 225, 227. 



Ex. 3. 

As{^"^- 



{ 
{ 



sm. 
sin. 
sin. 



: double area 
Area 



57° 31' 
63° 17' 
59° 12' 
15.65 

222.771 
111.3855 ch. 



A. C. 0.000000 
0.073890 
9.950968 
9.933973 
1.194514 
1.194514 
2.347859 
11 A., K., 22.17 P. 



Article 359. 

Ex. 2. Here ABC = 93° and BCD = 106°. 

As rad. A. C. 0.000000 

: sin. ABC 93° 9.999404 

( AB 12.47 1.095866 

•• \BC 11.43 1.058046 

2.153316 

A. C. 0.000000 

9.982842 
1.058046 
0.961985 

2.002873 



A. C. 0.000000 
9.512642 
1.095866 
0.961985 



: 1st quantity 


142.336 


As rad. 
: sin. C 
/BC 
'' ICD 


106° 
9.16 


: 2d quantity 


100.664 


As rad. 
: sin.B + C- 

/AB 
'''' I CD 


180° 19° 


: 3d quantity 


37.196 
100.664 
142.336 


2 ABCD = 
ABCD = 


280.196 
140.098 ch. 



1.570493 



= 14 A., R., 1.56 P. 



45 



p. 227.] 



KEY TO ALSOP'S SURVEYING. 



[Chap. V, 



Ex. 3. Here B = 48° 30' and C = 120° 30'. 



As rad. 
: sin. B 
f AB 
••* IBC 

: 1st quantity 



48° 30' 
8.63 
9.2T 

59.917 



A.C. 0.000000 
9.874456 
0.936011 

0.967080 

1.777547 



As rad. 
: sin. C 
TBC 

: 2d quantity 



120° 30' 

11.23 

89.697 



A.C. 0.000000 
9.935320 

0.967080 
1.050380 



1.952780 



Area = 



As rad. 
: sin. 180° - (B H- C) 11° 

TAB 
'• ICD 

: 3d quantity 18.492 

59.917 + 89.697 •- 18.492 



A. C. 0.000000 
9.280599 
0.936011 
1.050380 



1.266990 



131.122 



= 65.566 eh. 



= 6 A., 2E., 9P. 



46 



Chap. V.] 



COMPASS SURVEYING. 



[P. 235. 



CO 
CO 

O 

I— I 



CO 



o 

w O 

$ o 
S o 

-^ o 

02 O 


O 

to 
o 


to 

co" 

o 

CM 


o 
oo 

oo 

o 
1 




in 

125 






1 






CO 
I— ( 
t^ 

o 

-^" 

o 


£ 

•l-l 


o 

o 

o 
o 


r-H 

CO 
.1—1 


OS 

Co' 


cq 

CO 


CM 

cvi 

CO 


I— 1 

o 

I— 1 




1 1 


1 — 1 
CO 
1—1 


2 --J OC 1 (M O 
^ CO O GO ■ 




^ 








to 1 CO 

00 ^ 

CO -^ 

T— 1 


'^ CO 


o 

C5 


O O 1 
CO Ci 1 

rH CO 


O ^ -* 1 O 
O -«:f 0-3 Cs 

^ O ci CO '^ 




G5 1 
» CO 


1 1 o 

1 1 1>- 

1 a6 
1 I— 1 






rH ^-1 O O rH 
1 1 1 




o 


O 


o 


o 


o 


1— 1 




^ 










o 

oo 

o 


to 


f4 


CO 




I— 1 

CO 

!— J 


o 

Oi 
CO 






W2 




O 

o 
o 


r— ( 


CO 

co' 


o , 




12^ 


00 










00 


S 


1>^ 


o 
o 

1— 1 


O 
lO 


CO 

as 


oo 


CO 

CO 


9 


f4 

V 

o 

T— 1 

o 


V 

lO 
(M 



o 

CO 


lO 

to 


to 




to 
CM 


lO 

CO 

o 

CO 


M 


1— 1 


(M 


1 

CO 


Tf 


to CO 



to o 


o 


CO I— 1 


1 — 1 


^ r- 


t^ 


C5 O 


<M 


Ol ^ 


oo 


to o 


M< 


CO (M 


^ 



c<i 

CO 

cq 
<m' 

CO 



^ II 



i-H 

cd 
pi 






47 



p. 235.] 



KEY TO ALSOP'S SURVEYING. 



[C11A.P. V 








1 
1 


CO 

00 
as 

OG 

1— ( 


1 


OS 
OS 
I— ( 

to* 



r— 1 






I— I 

to 




to 

I-H 

0* 

to 


Is 

to 

06 

I— 1 


!5 






as 

CM 

CO 


c4 

OS 










1 




CO 


I— ( 
CO 

1>^ 


CO 



00 

i-H 


I— 1 

rH 

to 


si 

cd 






CO 






OS 


OS 


to' 


p 






-* CO 

rH 


CO 

§1 i 


^ 


00 
CO 

CO* 

rH 


1'—' 




10 






CO 

r— 1 


B 




10 

as 



I— ( 


as 
IS 


CO 
!>; 

rH 






tzi 








CO 

to 
^* 


to 

CO 

0* 
1—1 


b- 

rH 
1—1 


CO 

CO 


Z 


t— 1 


I— ( 
00 


OS 

to 

(M* 

I— 1 










i^ ! ^ 




















0^ 


1—1 








T-{ 













^ 


1 r-l I 10 

CO 1 ca 1 

CO 1 T-H 1 1 C4 
rH 1 1 1 |,-l 




CO 
t-H 


1 "^ 

^ 

1-H 1 1 


t- 1 CO I 
OS 1 b-. j 

CO* rH 1 

-T-{ I— ( 1 




DQ 


1 t- 10 

CO 

1 '^ 1 
1 1 rH 


CD 

rH 
I— ( 


CO 

!>; 

CO* 


CO ^H 1 
CO 00 1 UO 

^ 2^ "^ ! Sj 








. ( r-H 10 to GO 

-g rj" -:^ 10 00 
p oc CO cm' 1 t-^ 1 


CO 
CO 

CO* 

I-H 


OS 
CO 


to 

c 

03 


'^ 


CO 
-cf 


V 



CO 


»o 


CO 


to 


to 

to 


It- 


s 



to 



V 

to 


00 


CO 


CO 


1 rH <M CO 


'C^ 


to 


CO 


t- 



'Thi as 


to 


C<J Ol 


as 


CO CO 


CN 





CO 


-rf C<1 


<N 


CI OS 


CO 


rt^ 


CO 


I— 1 


1-H 




- — » 



<M 






CO 



48 



Chap. V.] 



COMPASS SURVEYING. 



[P. 235. 









^ 



o 

CO 
CN 
Gvi 

CO 
lO 
CO 



CO 





o 




o 






-* 




]— 1 






'^ 




!— 1 






iO 




^ 






o 




1^ 






I—I 










I— 1 









o 






o 

CO 



o t- ! cq 

CO I -^1 o 

r-~ I rH I CO 

O I uo I CO 



c\i I Ti^ 



^ ! 



'^ 



--0 

to 



o 

T— I 

lO 



CO 

co' 



oo 
CO 



^ 






CO 
CO 



o 



lO 
CO 

o 

CM 



o 






C-1 



(M 



(N cq 



cq 



cq 



CM 



lO 



CO 
! o6 



oo 

T— ( 

lO 



o 

CO 

00* 
CM 



o 

CM 



CM 

CO 

CO 
<M 



I "*. 
] OO* 
I CM 



>o 

lO 

o 

CO 



-t^ 



oo 

CM 



o 

o 

o 



o 

CO* 
CM 



CM 

oo 



lO 
CM 



O J.-I 

<^^ I CO 



^ 



oo 
o 

cq 



I ^ I ^ 
'-0 CO 

no T— I 



o I ^ 

CO CO 



"^ h» 12; \^ 



t^ 



02 



fci U^ 



1^ 



^ hzi 



cq 



O I t^ 00 



00 


o 


OO 


CX; 


<» 


CO 


CM 


T— 1 


r— 1 


CO 


t- 


CO 


Oi 


CO 


o 


oo 


o 


CM 


Ci 


C75 


O 


O 




O 



lO 

OO 
oo 

CO 

oo 



CM 



lO 

o 



lO 
CM 



49 



p. 235.] 



KEY TO ALSOP'S SURVEYING. 



[Chap. V. 



X 



to 

2 

•< 
m 




o 


O 

CO 
CO 


I— J 
o 
1—1 


o 
>o 

Ci 

CO 

o 




to 

1—1 

oi 


CO 

00 

Ol 

r-H 

o" 
o 

04 


Ol 

Ol 
Ol 

CO 

Ol 

CO 




S3 












O 
O 

>o 

o 








o 
to 
to 

-^* 


CO 

i-i 
.2 




oo' 

r— 1 


o 

CO 


1—1 


oo 

o 

Co' 


00 


to 
oq 

o 
oq 


to' 

CO 


oq 

CO 
• 

oo 


o 

CO 

1—1 






o 

CO 

1—1 


(M 

od 
.— 1 


o 
oo 

1—1 












to 


oq 

T— i 










to 
o6 


CO 

o 


o 

oq 

1—1 


oo 

Ol 

I— 1 


oq 

to 

to* 

1—1 








^ 


00 
CO 


oo 

r— 1 
















oq 

o 

oc* 


H 








lO 


GO 

o 
1— 1 


T— 1 

oq 
I— J 


o 

o 

1 — 1 


to 

lO* 






QQ 


o 
oq 












1—1 

CO 




oq 

CO 

I— 1 




^ 




I— 1 

• 
1—1 


Co" 
1—1 

r— 1 




00* 


o 
oo 

'^' 








CO 




1— ( 
I— 1 


1—1 


CO" 
I— 1 

cm' 

I— 1 


r— 1 

C5 


oq 

oq 

co' 

T— 1 


uo 

o 
^' 


o 
oo 

o ' 
1—1 


oq 
oq 


o 

CO 

•^* 

1—1 


CO 

o 

00 


.S 

<S 

13 
o 

□ 

03 

o 


V 

O 

CO 

CM 


o 

oo 


o 


p4 

V 

i-O 

r— 1 
O 

cr> 


O 

o 

1—1 
to 


V 

O 
1—1 

o 

I— 1 


o 

o 

00 

o 


V 

to 
I— 1 
o 
O 
CO 

m 


o 
to 
o 

CO 


to 

CO 

o 

1^ 

00 


c ■ 
o 


N 

o 
'J2 


o 

-^ 

o 


"A 

V 

O 
CM 

o 

"it) 
CO 


3 

V 

1-.0 

oq 



Ol 

00 
m 


3 

o 

oo 


V 

O 

CO 

o 


'A 

V 

o 

Ol 

o 

CO 

CO 


V 

to 

to 


o 
1—1 
o 


V 

to 

—1 
o 

oq 

to 




1 


(M CO 


1 
-^ O o 


\r~ 


00 


05 


o 

1 — 1 







to ^ 




S5 <==> 


Ci 


-* 


a. 


oc (M 


O 


CO 


CO 


to r-l 


-^ 


1^ 


Ci 


1—1 o 


to 


Ol 


o 


li ^ 


o 


CO 


CO 


O Ttl 


-^ 




1— i 


1— ( 




1~— 




1-1 


:X 


to 


. 



o 
oq 



o 



I-- 



oq -^ 



o 
o 
o 
o 
o 
o 



Ol to 

o o 

CO t— 

o oo 

O Ci 



O Oi O 



oo 








r^ 








^■i 


CO 


C-3 


CO 




o 


o 


Ci 


CO 


o 


o 


Ci 


CO 


C-5 


CO 




to 


(-1 


oo 




o 


o 


Ci 


"^. 


o 


i^ 


o 


CO 









oq 



P^ 



cq 

to 

'*' 



u 

r— < * -1^ • 

^ 3D 33 -+J 



to 

o 
00 

Ci 

o o o* 



to Jf^ 

od Oi 









O 








TJ 
















(/i 








rC! 








T^ 


P 


^ 




-4-3 

05 


QQ 


r-l 


r3 


'TS 



60 



Chap. V.] 



COMPASS SURVEYING. 



[P. 235. 



Tl2 

< 








GC 


O CO 

•CO o 

-^ Ci 

o o 
"^ ci 


1—1 

GO 

-^ 
o 

to' 

Ci 


< 




oo 
o 
oo 
CI 

o 


CI 

o 

r— 1 

o 

UO 






oo 

oo 
o 

CO* 

CO 

o 


CO 

o 

GO 

d 

GO 




o -^ 

CI (M 


o 
CO 


O Ci t^ 
-*_ CO iC 

-*' OO CI 

rH CI j ^ 


CO 
ci 


I— 1 
Ci* 

cq 














r— 1 


O Ci oo 

CI -*. o 

O ^ CO 
CI .— 1 r-H 


o 

'2. 






— ^ t^ 1^ 

O O <M 

o o -^ 

(N CM 










cq 




^ 






o 


Ci 

o 

r-l 


1—1 

rH 

oc 


CO 
CO 

o* 


o 

CO 






ITS <M 

~" I— 1 






1 






o 

CI 


m 








to 




CI o 

I— 1 rH 




1^ 




Tt^ oo 
^. ^. 

CO Ol 






1—1 
CI 

CO* 








s 


O 

TO 


■OO 
CI 


CO 
CO* 


o 
1— 1 


00 


CO 

cq 


o 
o 

-^' 

1—1 






.s 

OJ 

P5 

<B 

a 


o 

CO 

CO 


o 

O 

125 


o 

I— 1 


Ci 

z6 


o 

to 
oo 


o 
CO 


o 

o 

CO 

y5 


o 

CO 


o 

rH 
GO 


to 

a 

ci 

o 


3 

o 

r/2 


o 

co'-f 

OO 

CO 




I— 1 


o 

CO 

rn 


o 

oo 


o 
Hc-1 

>o 


o 

oo 


o 

I— 1 

czi 


o 

Ci 




1—1 


1 1 1 1 

CI 1 CO Tti lO o 


t— 


oo 


Ci 









o 


o 


cq 


l-l -^t* 


TJH T^l 


o 


xf 


t^ 


oc 


lO 


1^ 


CO 


-t^ 


r— ' 


■oc 


Ci 


t- 


^ 


to 


cl 


r—^ 


GO 


-* 


cf 

Ci 


I— 1 


rH 


a; 




C-1 


I— 1 






' — ■ 


. 


Ph 



c^ ^ 



o 
o 
o 
o 
o 
o 



Cl 


Oi 


CO 


.-H 


CO 


<-> 


'^ 


CI 


C^i 


1—1 


r-H 


CO 


Ci 


r-H 











cq 






- 




CO 






^ 










a 






CO 




a:> 






oo 




,^ 






^ 


cJ 


o 


to 

.»H 


I-H 




fH 


a 


'Ci 




00 




.. 




CO 

oo 


<1 




•• 


• ■ 


o 










-^ 










t^ 










o 


O 


O 


Ci 


<-Ti 


o 


GO 


o 


CO 


rH 


-^ 


CO 


o 


c.O 


o 




to 


o 


o 


CI 




o 


o 


o 


.— 1 


1^ 


o 


o 


•CO 


CO 


o 


o 


o 

t— 1 


i-H* 


l—J 


o* 


« 








'^J^ 











<lO 1—1 

cq to 

o o 

Cl cq 



d 






51 



p. 237.] 



KEY TO ALSOP'S SURVEYING. 



[Chap. V. 



CO 

I— t 



X 



eS 
< 




CM 

00 

00 


r-l 
I-- 

>-( 
CO CO 

r— 1 


r-H 

Ol 

(M 

GO 

CM 
CM 


i 
< 


! 


\ 

1 




is* 




CO 
1— 1 


p4 


QO 
1 '"' 


p4 

CO 
CXD 








I t- 

CO 1 -* 

r-H 0* 
rH (M 1 


d 1 CO t- 




CM 


1 1 


Tf CO 
CO rH 

•CO 1 1>^ 

rH 1 


w 


CO I-^ 

1 

l-H 1 CM* 

,-1 1 


1 


1'^ 


OJ 


1 

1 T-^ 






a' - 


1 
1 


1 '-' 
1 ^ 

rH 


-* 

(M' 








r-H 


i—t 


T^ 


1^ ^ 


(M 


rH 1 rH rH 

1 * 1 


^' 


CO 
CO 
• 
CO 
r-K 


T-H 




w 



i-H c i 


1 




CO 


00 

1 T-H 


l-H 


1^ 


1 









(M 




s 


CO 

I— ' 


I— 1 


CO 

Co' 
rH 


^. 
CO* 
rH 





10 

bo 
C 


1 
' 

l^ 1 




CM 




CO 


"A 



CO 


[» - j 


CO 


■^ 


lO 



-* t- 



CO 













to rt 


^ 





rH , 


G<1 


00 


Cj 








-^ 


r~ 


0-2 





f>J 





'O 


r-\ 


CO 


uo 


i- 








c:. 


Tt^ 


CO 







CO 




r^ . 



cq <tj 



P5 



o 

o 

r-l 

o 

p 





Inter. 
Dist. 


Sum of 
Offsets. 


Areas. 




4.80 


.75 


3.6000 , 




2.59 


1.72 


4.4548 




1.57 


1.69 


2.6533 


> 


1.58 


1.27 


2.0066 


-a 


.78 


.55 


.4290 


< 


1.01 


.50 


.5050 




1.72 


.95 


1.6340 




1.92 


1.32 


2.5344 




2.21 


1.22 


2.6962 1 




1.77 


.98 


1.7346 




3.26 


.63 


2.0538 j 



24.3017 



2.95 
1.57 

.99 



.75 

1.50 

.75 



2.2125 

2.3550 

.7425 



_5.3100 

is".99iT 



52 



Chap. V.] 



COMPASS SURVEYING. 



[P. 23^ 







o 
o 

t^ 

CO 


O 
t— 

o 

i—i 






O 
CO 

<M_ 

CO* 
r-l 


< 
125 


O 

-^ 

t-; 

CO 


o 
oo 

00 
CO 

oo 


o 
CO 

CO 




to 
u 

[a 
9 


O 
cc 

i-H 


-^ 00 
t- CO 

o6 o 

C<1 (M 


o 


to 
cq 

CO* 


p4 

cq' 






P 


-* o 

O 00 
I— 1 


as 
oo' 




P 
P 


1 O 00 Tfi 1 o 

CO 1-M oj 1 oq 

cm' ci I o od 

1 1 i-f 


^ 


^ 1 o 
o t- 

ci O 


i 


e4 






00 


to 


OO 


o o 

CO O 

cq o 


:/i 


t— 

o 




1"^ 
C5 


ai 


o 


O 
CO 


125 


O 1 rJH 1 
oo 1 t- 1 

j oo' 
1 — 1 








t— 1 


I— 1 


o 


o 


rH 


o 


o 


a^' 


rH 


i—i 


o 


1 

o o o 


o 






^ 


CO lO 
O b-; 

C5 O' 

I— 1 


! - ! 
i 1 


H 






oq 1 Tt^ vo o lo 

00 1 lO O CO 1 uo 

.-H 1 * j 00 j oi CO* 


CO 




o o o o 

o t^ t^ "^ 


^ 


1 00 
1 i-H 


^ ! 1 
i 1 i 


p 


oo 1 lO 
(N 1 CO 

i-H r-I 
i-H fM 


O lO C5 

as o 00 

rA rA O^ 


O 1 C<1 

(?q* aj 


in 
bO 



1 


O 
1— 1 
o 

CO 


s 
o 


V 

00 
o 

o 


f4 

s 

00 

o 

o 

CO 


V 

o o 

T— 1 CO 

o t- 


3* 

o 

00 
G^ 
o 

C/2 




t— ( 


Q^ 


CO 


•^ 


o 


o 


t^ 











o 


o 


CO (^q 


00 . -*l ^ 


CO '^t' 


(N 


T— 1 I— 1 


fN 


CO 


VO 


1— ( 


lo oo 


CO 


1— 1 


CO 


to 


G<J >-l 


'^ 


J^- 


00 

cq 


o 


ai <M* 


I— 1 


lO 


1—1 


ai 1—1 


I— 1 




cq 


rM 


CM 


CO 


rH 





Cl 



i:^ 



5h^ 
o 



o 
ft 



oo 
OS 



Inter. 
Dist. 


Sum of 
Offsets. 


Areas. 


.11 

.94 


.44 
.79 


.0484 
.7426 

.8100 
3.3024 
4.0500 
2.7218 

.4800 
.0560 

12.2112 

300 


1.08 
1.92 
2.50 
4.39 


.75 
1.72 
1.62 

.62 


1.50 
.40 


.32 
.14 


5SS .50 


.06 




12.1812 
53 



p. 242.] 



KEY TO ALSOP'S SUllVEYING. 



[Chap. V, 











rH 


^ 


r- 


t^ 






to 




O 


O 


to 


OO 






oi 




o 


CN 


I— 1 


to 






a> 

Si 




oo 


O 


CO 


(M 






<! 




c<i 


o 


CO 


-f 






ai 




o 

CM 


CM 


CO 


o 
I— 1 






03 
















CD 
















b 
















<1 
















'^ 














in 

Si 




^ 


^ 


^ 


H 






'Hi 




T— ( 


t- 


t— 


CO 






i3 




b- 


CO 


!>; 


t-; 






-3 




-*' 


ai 


oi 


(M 






g 




(M 


(M 




r-^ 




1 




W.D.D. 


CO 


1— ( 


CO 










t^ 


1:^ 


OS 










oi 


'^' 


^' 










T-i 


CM 










P 
P 








O 


O 












Oi 


iq 












cji 


c^i 




to 


f4 








1— 1 


(M 






^ 


1^ 










a 




lO 


,—1 












^ 


1>^ 

T— 1 


1>^ 










> 








I— ^ 


as 


--^ 




;-i 








(M 


o 


O 


. 


P3 


W 






c4 


Ir-^ 


'^' 


CO 


GQ 










I— 1 




CD 




>o 








Ci 


CO 




02 


lO 








1— ( 


o 


CO 
rH - 




1— I 
1—1 








00 


■7^ 






I— 1 


(M 


O 




CP 






(M 


1—1 


^. 




•iH 


• i—i 


;2q 




o6 


cc 


OO* 




•^ 




< 


&>-■ 
















o'^ 




























c4 




rH 












M 








•^ 


ir- 










P^ 


^' 


to 


t—i 












t-^ 


t-^ 














I— 1 


















rH 


Ci- 


-H 












CN 


co 


OO 






«■ 






c4 


1—1 


'*' 






CD 








Oi 








to 








rH 






ai 


1—5 

1— 1 








00 








T— 1 


CM 


O 












CN 


rH 


"*. 








^ 




cx). 


cx5 


co' 






1 






o 


o 


C-J 


1— 1 


O 






aj 


o 


'Ci 


^. 


o 


to 






P 


r-^ 


o 


od 


o6 


CJ 








<M 


1—1 




rH 








^ 


^ 


p4 


W 


p4 






w 


V 


V 


V 


N 


V 






M 


t^ 


t^ 


to 


b- 


'^ 






■ "r* 


CO 




1—1 




(M 






t4 


o 


o 


o 


O 


o 






4) 


O 


T— ( 


lO 


Ci 


O 






« 


lO 


-* 


1— 1 


I- 


CO 




^ 




•J2 


zn 


l^ 


1^ 


aj 




i 


I— ( 


(M 


(TO 


Tt< 


to 






OQ 













CO 
CO 

CO 
r-A 
OO 

to 



to 

OO 



as 

to o 

I— I OO 

tq CO 

|>I CO 

T^ CO 



o 


O 


CO '^ 


•^ Tf 


to 


(M 


1— 


O 


00 


to 


-h 


CO 


CM 


. 




p^ 



<1 



5a 
o 

o 






<1 





o 


^ 


on 


to 


b- 




o 


<M 


CO 


r^i 


a 




o 


^ 


CM 


CO 


CO 




o 


(Tl 


r^ 


O 


C_J 




o 


^ 


Oi 


(M 


CU 




o 


CX) 


r^ 


Oi 


lij 




o 


Oi 


o 


o 


I— ( 




o 










t— ( 












t^ 

Tt 


< 










CM 












I— 1 












t- 












'^ 












CM 












to 

CO 




to 

1 


CM 

CO 


OO 

CO 

oo' 


CO 
CO 
CO 

CO* 

OO 


t- 














54 



Chap. Y.] 



COMPASS SURVEYING. 



[P. 242. 



Offsets, 



! 

! Base. 


Dist. 


Offsets. 

.00 


Inter. 
Dist. 


Sum of 

Offsets. 


Areas. 




.00 










.20 


.15 


.20 


.15 


.0300 




1.85 


.25 


1.65 


.40 


.6600 




3.50 


.10 


1.65 


.35 


.5775 




5.85 


.82 


2.35 


.92 


2.1620 


AE 


6.27 
14.60 


- 


.42 


.82 


.3444 




15.00 


.60 


.40 


.60 


.2400 




16.60 


.25 


1.60 


.85 


1.3600 




18.90 


.05 


2.30 


.30 


.6900 




20.35 


.b^ 


1.45 


.60 


.8700 




21.00 




.%^ 


.bb 


.3575 




.00 


.00 








.45 


.95 


.45 


.95 


.4275 




1.50 


.15 


1.05 


1.10 


1.1550 




2.75 


.10 


1.25 


.25 


.3125 


BC 


4.75 


.85 


2.00 


.95 


1.9000 1 




6.75 


.70 


2.00 


1.55 


3.1000 




9.37 


.05 


2.62 


.75 


1.9650 




10.50 


.50 


1.13 


.^b 


.6215 ! 




10.90 




.40 


.50 


.2000 




.00 


.00 








.15- 


.57 


.15 


.57 


.0855 




2.25 


.40 


2.10 


.97 


2.0370 


CD 


4.00 


.00 


1.75 


.40 


.7000 




6.50 


.37 


2.50 


.37 


.9250 




7.85 


.90 


1.35 ! 


1.27 


1.7145 




8.42 




.57 


.90 


.5130 



22.9479 



00 



p. 242.] 



KEY TO ALSOP'S SURVEYING. 



[CflAP. V. 



Base. 


Dist. 


Offsets. 


Inter. 
Dlst. 


Sum of 
Offsets. 


1 i 

Areas. ' 




.00 


.00 








.b5 


.92 


.55 


.92 


.5060 


• 


2.75 


.25 


2.20 


1.17 


2.5740 




4.50 


.05 


1.75 


.30 


.5250 




6.50 


.20 


2.00 


.25 


.5000 


DE 


9.00 


.10 


2.50 


.30 


.7500 




11.50 


.50 


2.50 


.60 


1.5000 




13.50 


.50 


2.00 


1.00 


2.0000 




16.00 


.10 


2.50 


.60 


1.5000 




17.50 


.50 


1.50 


.60 


.9000 




18.01 




.51 


.50 


.2550 




.00 


.00 








.27 


.70 


.27 


.70 


.1890 




1.50 


.25 


1.23 


.95 


1.1685 




3.00 


.30 


1.50 


.55 


.8250 


EA 


4.00 


.00 


1.00 


.30 


.3000 




5.00 


.00 


1.00 


.00 


.0000 




7.50 


.70 


2.50 


.70 


1.7500 




9.00 


.20 


1.50 


.90 


1.3500 




9.50 




.50 


.20 


.1000 


. 


.00 


.00 










.20 


.90 


.20 


.90 


.1800 


GF 


2.50 


.00 


2.30 


.90 


2.0700 




4.75 


,65 


2.25 


.65 


1.4625 




6.27 
.00 


.00 


1.52 


.65 


.9880 








1.50 


.20 


1.50 


.20 


.3000 


FH 


3.00 


.15 


1.50 


.35 


.5250 




5.60 


.70 


2.60 


.85 


2.2100 




5.80 




.20 


.70 


.1400 



Brought over, 



56 



24.5680 
22.9479 

47.5159 



CHAPTER VII. 

LAYING OUT AND DIVIDING LAND. 





Article 378. 






Ex. 2. 




\ sin. A 


11.25 

73° 25' 


A. C. 8.948847 
" " 0.018451 


: 2 ABC 


70 ch. 


1.845098 


: : rad. 




10.000000 


: AC 


6.49 


0.812396 


Ex. 3. Here B = 


= 117° 15'; .-. 




As ^ . ^ 
{ sm. B 


16.37 
117° 15' 


A.C. 8.785951 
" " 0.051090 


: ABCD 


160 


2.204120 


: : rad. 




10.000000 



BD 10.99 1.041161 



Ex. 4. Here the given angle is 72° ; .-. 

14.37 A. C. 8.842543 



Aq 

' sin. 72° " " 0.021794 

: double area 200 2.301030 

: : rad. 10 .000000 

: other side 14.63 1.165367 



iii 



Pp. 256, 257.] 



KEY TO ALSOP'S SURVEYING. 



[Chap. VII. 



Article 380. 

Ex. 2. Here the angles are 102° 45', 45° 45', and 31° 30'. 



As 



102° 45' 
45° 45' 



siu. 

sin. 

rad. 

sin. 31° 30' 

: : 2 area 100 

: square of side 
Side 8.648 



A.C. 0.010843 

^' '' 0.144904 

10.000000 

9.718085 
2.000000 



2) 1.873832 
.936916 



As 



Ex. 3. Here A =-- 62° 16', B 

Then 

sin. A 62° 16' 

sin. B 48 
rad. 

sin. C 69° 44' 

: : 2 ABC 240 
: AB^ 

AB 18.50 



48°, C = 69° 44'. 



A. C. 0.052996 

" " 0.128927 

10.000000 

9.972245 

2.380211 



2)2.534379 
1.267189 



As sin. I 
: sin. M 

:: PM 
: PI 



Ex. 2. 

85° 

90 

3.25 

3.26 



Article 381. 

60 

^^ = Y.F2 = '■''''■ 



A.C. 0.001656 

10.000000 

0.511883 

0.513539 



PH = A D - IP = 4.32. 

DC = y/ FC^- FD^ = n/7.58 x 1.06 

= v/OMS = 2.84. 
AC = 7.58 + 2.84 = 10.42. 




58 



Chap. YII.] 



LAYIisG OUT AND DIVIDING LAND. [Pp. 258, 261. 



Ex. 3. Fig. 1T2. Surveying. 

As sin. I 88^ 15' 

: sin. PAI 42° 

: : PA 5.64 

: PI 3.776 



A, C. 0.000203 
9.825511 
0.751279 



0.576993 



As rad. 






: sin. PAL 




46° 15 


:: PA 






: PL 




4.074 


Given area 




35 


FE + FD =-- 


AD 


8.591 


FE - FD 




1.039 


DC 




2.99 


AD 




8.59 


AC = 




11.58 



A.C. 0.000000 
9.858756 
0.751279 

0.610035 
1.544068 



.934033 
.016616 

2) .950649 

.475324 





Article 383. 




Ex. 2. Fig. 175. 


Surveying. 






/sin. A 
^' I sin. B 


90° 




A.C. 0.000000 


71° 30' 




" " 0.023043 


( rad. 
• I sin. C 


18° 30' 




10.000000 
9.501476 


:; 2i\BCD 


140 




2.146128 


: AB2 - CD^ 


46.8432 




1.670647 


AB^ = 


161.7984 


10.72. 




CD = 


n/114.9552 = 




And 








As sin. E 


18° 30' 




A. C. 0.498524 


: sin. B 






9.976957 


: : ^B - CD 


2.00 




0.301030 


: ^D 


5.98 




0.776511 



59 



Pp. 261, 262.] IvEY 


TO ALSOP'S 


SURVEYING. 


[CUAP 


Ex. 3. Fig. 174. 


Surveying. 






. r sin. A 

As \ . T5 

i Sin. B 


90° 


A.C 


. 0.000000 


119° 30' 


a a 


0.060303 


f rad. 
• t sin. E 






10.000000 


29° 30' 




9.692339 


: : 2 ABC.D 


240 




2.380211 


: CD' - J\B2 


135.785 




2.132853 


AB^ = 


134.56 







CD = ^/270.345 = 16.44. 



As sin. E 
: sin. B 
:: CD - 


AB 


4.84 




A.C. 


, 0.307661 
9.939697 
0.684845 


: AD 




8.555 
Article 384 






0.932203 


Ex. 2. Fig. 176. 
= 87° 30'. Hence 
than AI. 


Surveying. 
K lies below 


He] 


re A = 91° an 
and AL is gr( 


Area 
AB 




100 eh. 

7.86 




A.C. 


2.000000 
9.104577 


AI 




12.72 






1.104577 


AI 

tan. EAI 
IK 




12.72 
2° 30' 
M5 






1.104577 

8.640093 
-1.744670 


And AI 
tan lAF 




1° 






1.104577 
8.241921 


IF 




.222 






-1.346498 



Whence FK = IK - IF = .333. 
And EF = AB - FK = 7.527. 



60 



Chap. VII.J LAYING 


OUT AND DIVIDING 


LAND. [Pp. 262, 266. 


Then 






As 2 EF 


15.054 


A.C. 8.822348 


: AI 


12.72 


1.104577 


:: FE 


.333 


-1.522444 


: IL 


.281 


-1.449369 


AL = AI - IL 


= 13.001. 
Article 385. 




Ex. 2. Fig. 178. 


Surveying. 




As sin. E 


18° 30^ 


A. C. 0.498524 


: sin. B 


71° 30' 


9.976957 


:: AB 


12.72 


1.104487 


: AE 


38.016 


1.579968 


i\B 


12.72 


1.104487 


sin. A 


90° 


10.000000 


2 ABE 


483.565 


2.684455 


2ABCD 


140 




2CDE 


343.565 




, r sin. E 

As < . -p. 

{ sm. D 




A. C. 0.498524 


116° 47' 


'' " 0.049286 


( rad. 
• t sin. C 




10.000000 


81° 43' 


9.995446 


:: 2ECD 


343.565 


2.536009 


: ED^ 




2)3.079265 


ED 


34.644 


1.539632 


AE 


38.016 




AD 


3.372 




And 






As sin. C 


81° 43' 


A.C. 0.004554 


: sin. E 


18° 30' 


9.501476 


:: ED 




1.539632 


: CD 


11.11 


1.045662 



HI 



Pp. 2G6, 268.] 



KEY TO ALSOP'S SURVEYING. 



[Chap. VII. 



Ex. 3. Here A = 106° 56\ B = 111° 4', E 
D each 71°, and bear, of CD K 30° 43^ E. 



38°, C and 



As sin. E 


38° 


A. 


C. 0.210658 


: sin. B 


111° 4' 




9.969957 


:: AB 


15.62 




1.193681 


: AE 


23.675 




1.374296 


AB 






1.193681 


sin. A 


106° 56' 




9.980750 


2 ABE 


353.775 




2.548727 


2ABCD 


200 






2CDE 


553,775 






And since EC 


= ED, we shall have 






As sin. E 




A. C. 0.210658 


: 2ECD 


553.775 




2.743333 


: : rad. 






10.000000 


: ED2 






2)2.953991 


ED 


29.991 




1.476995 


EA 


23.675 






AD 


6.316 







Article 386. 

Ex. 2. Produce AF and BC to meet in E. 
To find AB, ABE, and EF. 

11° A. C 

71° 16' 



As sin. E 

: sin. B 

:: AB 

: AE 

AB 

sin. A 

2 ABE 

ABE 

ABCD 

ECD 

ECD 
PI 

EF 

62 



15.23 



45.59 



0.719401 
9.976361 

1.182700 



75.59 


1.878462 




1.182700 


97° 44' 


9.996032 


1140.76 


3.057194 


570.38 • 




150 


' 


420.38 




420.38 


2.623642 


9.22 


0.964731 



1.658911 



G-.-- 




Chap. YIL] LAYIXG OUT AND DIVIDING LAND. [P. 268. 

To find PR, AR, and BR. 

As siu. R 82° 16' A. C. 0.003968 

: md. 10.000000 

: : PQ 10.55 1.023252 



DG 



: PR 




10.647 


1.027220 


As sin. R 




82° 16' 


A. C. 0.003968 


: rad. 






10.000000 


:: AS 




9.22 


0.964731 


: AR 




' 9.305 


0.968699 


AR 




15.23 




BR 




5.925 




3 find HR and HP. 


- 




As sin. E 




11° 


A. C. 0.719401 


: sin. B 




Tl° 16' 


9.976361 


:: BR 




5.925 


0.772668 


: HR 




29.407 


1.468450 


PR 




10.647 




HP 




18.76 




+ FG = EF 





45.59 


1.658911 


- FG = EF ■ 


-2HP 


= 8.07 


0.906874 




- 




2)2.565785 






19.18. 


1.282892 



FD 

"Whence AD = EA - (EF + FD) = 10.82. 



G3 



Pp. 269,273.] 



KEY TO ALSOP'S SURVEYING. 



[CllAl-. VII. 



Article 387. 

Ex. 1. Fig. 180. Surveying. 
To find AE, ABE, and AD.^' 



As sin. E 


30° 30' 


A. C. 0.294531 


: sin. B 


44° 10' 


9.843076 


:: AB 


16.75 


1.224015 


: AE 


22.99 


1.361622 


AB 




1.224015 


sin. A 


105° 20' 


9.984259 


2 ABE 


371.446 


2.569896 


2 ABCD 


300 




Then 2 ECD 


671.446 




And 






As sin. E 




A. C. 0.294531 


: rad. 




10.000000 


: : 2 ECD 




2.827011 


: ED^ 




2)3.121542 


ED 


36.37 


1.560771 


AE 


22.99 




AD 


13.38 





Bear, of DC, K llj W. 



Article 388. 



St. 

T 

T 

T 


Beariogs. 


Ch. Bear. 


Dist. 
5.00 


N. 


S. 


E. 
.11 


W. 


E.D.D. 


W.D.D. 


Mult. 


' "1 
S. Arena. 

77.6500 


N. 27^0 W. 


N. li°E. 
N.29i0W. 


15.53 


15.53 W. 


N. 58° W. 


9.53 


8.29 





9.08 


4.69 


4.58 


20.11 W. 


106.7119 


N.42iOE. 


N.71° E. 


9.60 
14.00 


3.13 
^ _ 




4.39 




15.72 W. 


49.2036 
38.1150 


S. 81i°E. 


S. 52|OE. 


8.47 


11.14 




20.22 




4.50 E. 


S. 28iOE. 


South. 


(10.76) 





' 


11.14 




15.04 E. 






(15.64)1 




15.64 


00.00 E. 





20.33 



331.6805 
500 

168.3195 



Lat. of 5tli side = 



168.3195 
' 15.64 



= 10.76. 



64 



Chap. VII.] LAYING OUT AND DIVIDING LAND. 



[P. 275. 






N. 


S. 


E. 


W. 

1 


E.D.D. 


W.D.D. 
1.45 


Mult. 


N. Areas. 


S. Areas. 


AF 


.32 


14.83 








Fa 


5.10 


12.31 


27.14 


27.14 E. 


138.4140 




GH 




14.82 
1L78 


7.63 




19.94 


10.86 


47.08 E. 




697.7256 , 


HB 


(21.82) 


18.49 
(16.28) 


36.22 E. 




426.6716 


BA 






34.77 


1.45 E. 


31.6390 


i 



26.92 26.92 34.77 


34.77 


170.0630 1124.3972 
170.0530 

2 AFGHB 954.3442 
2 given area 1100. 






2 ABCD 145.6558 


As Lat. AB 


21.82 


A. C. 8.661145 


: Dep. 


16.28 


1.211654 


: : rad. 




10.000000 



tan. bearing 36° 44' 



9.872799 



As COS. bearing 

: rad. 
: : Lat. 

: Dist. AB 27.23 



A.C. 0.096131 

1.338855 
10.000000 

1.434986 



65 



p. 275.] KEY TO ALSOP'S SURVEYING. [Chap. VII. 

Produce AM and BI to meet in E. Then 

As sin. E 34° A. C. 0.252438 

: sin. B 70° 16' 9.973716 

: : AB • 1.434986 

: AE 45.839 1.661140 

AB 1.434986 

sin. A 75° 44' 9.986395 

2 ABE 1209.27 3.082521 
2ABCD 145.66 

2CDE 

sin. E 



As ^ . -p^ 

sm. D 

( rad. 
\ sin. C 
: : 2 CDE 

: DE^ 
DE 
EA 

DA 2.963 

MD = MA - DA = 9.207 ch. 



66 



1063.61 






A. C. 0.252438 


104° 


" " 0.015056 




10.000000 


110° 


9.970152 


1063.61 


3.026782 




2)3.264428 


42.876 


1.632214 


45.839 





Chap. VII.] 



LAYING OUT AND DIVIDING LAND. 



[P. 278. 



Article 391. 

Ex. 2. Fig. 182. Surveying. The division line will 
fall on the fourth side. 





CO 

CO 
CO 

CO 
CO 

CO 


117.3205 










CO 

CO 

OS 
CO 


CO 


CO ir- cq 

CO tH lO 
O 'T O 
lO CS CO 
t-1 -rl^ r-^ 
T^l ^H 0> 

1—1 cq 


- 


1— ( 
QO 

CO 


C<1 


CO 
I— 1 


OS 


p4 h h 

CO OS CO 

CO CO lO 

^ 1-J c4 

—1 C<I 


OS 

CO 

CO 




1—1 

CO 
CO 








OS 

o 

CO 

CO 




oi 


O 1 CO 


cq CO 
Cs 1-- 

tA CD 

1—1 


crs 

r-l 


^-1 
o 

CO 

T— 1 




^ 


















OS 
o 

CO 

CO 


p4 


50 


CO 


CO 


00 1-^ 
O lO 

oi CO 
1—1 


ai 






o 


CO 

o 








■CO 


so 

T— 1 

CO 
1—1 


N. 

8.83 
4.15 




CO CO -H 
C<J O OS 

o o CO 
I— 1 1— ( 






Dist. 
9.00 
12.00 
10.85 
10.62 
13.70 
10.30 


o 
c^ 

CO 
1—1 








1—1 
1—1 

12^ 




05 (» 

125 m 


o 

lO 1— I 




o 
w. 




50 

1 
s 


o 


f4 



«o 




1— < 




CO 

1—1 


o 





w 




CO 
CO 

oi 




s 


2 


- 


r-H 


CO 


1—1 


cq 


CO 


^ 


ft 



■t^ CO 

(M CO 

M CO 

CO r-l 

-t-i 1-; 



1 V y 




1 OS 
OS 

CO t- 
CO C-5 


OS lO 
CO ^ 
CO CO 


O CO 


CO CO 


-— N 





<=> oo o 

-^ CO O 

o i^ <o 

Ttl O CD 

J>. lO O 

CO rH O 

O 



CO 




■^ 




CO 


o 


OS 


CfJ 




T-l 



?t5 « -^ 

<5 •• .. 



OS 


o 


o 


OS 


^ 


lO 


CO 


C<l 



O 1-. I r-i 

d 



> 
















^ 


^ 






'^ 


o 






1—1 


I— 1 


CO 


1-1 


o 


o 


I— 1 


->* 


CO 


CO 


00 


o 


CO 


o 


r-l 


""^l 





bo 




n 






© 


£_( 


a 


ci 




Ol 


""* 


r^ 




CD 

o 
o 



. 285-288.] ] 


lEY T 


ALSOP'S SUR\ 


^ey; 


[NG. 


[Chap. VII. 






Article 399. 








Ex. 2. 












As AEC 




58 ch. 




A. 


C. 8.236572 


: ADE 




30 " 






0.477121 


(AVi 




12.25 






1.088136 


'' JAC 




10.42 






1.017868 


: AD.AE 








1.819697 


AE 




8.50 






0.929419 


AD 




7.77 . 






.890278 


Ex. 2. 




Article 400. 








As 5 : 2 : : 


AC^ 


(400) : AD2 = 


160, 




.-.AD = 


^/ 160 = 12.65. 












Article 401. 








Here, CF bein 


[g perpendicular to AK 


, we have 


As rad. 








A. 


C. 0.000000 


: COS. A. 




39° 45' 






9.885837 


: : AC 




13.24 






1.121888 


: AF 










1.007725 


AG 

AD 




6.365 
8.049 






.803798 
2)1.811523 
.905761 ' 



68 



Chap. VII.] 



LAYING OUT AND DIVIDING LAND. 



[P. 290. 



Article 402. 



Ex. 2. To find AE and 
EP. . 



As sin. AEP 




111° 30' 


: sin. APE 




33° 45' 


:: AP 




5.90 


: AE 




3.523 


And 






As sin. AEP 






: sin. Pi\E 




34° 45' 


: : AP 






: PE 




3.614 


To find AG. 






As AE 






: AF 




5.935 


:: AD 




4.4 


: Aa = KI + IG 


= 7.412 


2EP 




7.228 


El-IG 




.184 




KG 


= 1.168 



Fig. 18. 




A. C. 0.031322 
9.744739 

0.770852 



0.546913 



A. C. 0.031322 

9.755872 
0.770852 

0.558046 



A. C. 9.453087 
0.773421 
0.643453 



0.869961 



- 1.264818 

2 )0.134779 

0.067389 

And AK = AG ± KG = 7.412 ± 1.168 = 6.244 or 8.58. 

69 



Pp. 291,293.] KEY TO ALSOP'S SURVEYING. 



[Chap. VII. 



Article 403. 

Ex. 2 Fig. 19. 

DF = 1 CD = 3.96 and AE = i AB = 4.97T. 
To find PM and AM. 

As sin. M 83° 50' A. C. 0.002520 



Fig. L 



: sin. PAM 


54° 50' 


9.912477 




M V 


: : AP 


6.20 


0.792392 


A 


'\' i 


: PM 


5.098 


0.707389 


H 


\\iL. 


As sin. M 


A.C 


. 0.002520 


E 


--^'^. 


: sin. APM 


41° 20' 


9.819832 






: : AP 




0.792392 






: AM 


4.119 


0.614744 




^ 


D find ML, LP, 


and FG. 




-• B 


^ 


As AD 


5.18 


A.C. 


9.285670 


: AM 






0.614744 


: : AE - DF 


1.017 




0,007321 


: AE-ML 
AE 


.809 . 
4.977 


— 


■1.907735 


ML 


4.168 









LP = MP -ML = .93 

As AM — J AD 1.529 

: JAD 2.59 

: : PL .93 

: FG = EH 1.575 



A. C. 9.815593 
0.413300 

-1.968483 



0.197376 



whence 
DG = DF + FG = 5.535, and AH = AE - EH = 3.402. 



Ex. 2. 
EF=J( 



Article 404. 

4BC2 + 5 . ADS / /426.0096 + 154.0125^ 



9 



■) = J0 



9 



= v/ 64.4469 = 8.03. 



70 



Chap. VII.] LAYING OUT AND DIVIDING LAND. [Pp. 293, 294. 

And as 

BC - AD (4.77) : FE - AD (2.48) : : AB (6.47) : AE = 3.36. 



Ex. 2. 

To find AG and BG. 



Article 405. 



As sin. G 
: sin. B 

: : AB 
: AG 

As sin. G 
; sin. A 

: : AB 
: GB 



34° 

62° 30' 
19.55 
31.01 

83° 30' 
34.74 



A. C. 0.252438 
9.947929 
1.291147 
1.491514 H 



A. C. 0.252438 c 
9.997199 
1.291147 
1.540784 



Fiff. 20. 



To find GH and BI. 




AsGA 


31.01 


A. C. 8.508486 


: GD 


18.76 


1.273233 


: : GO 


15.82 


1.199206 


: GH 


9.57 


1.980925 



BI = f (GB - GH) = 13.98 ; GI = GB - BI = 20.76. 



To find GF and BF. 



AsGP 


25.01 


A.( 


1 8.601886 


: GA 


31.01 




1.491514 


: : GI 


20.76 




1.317227 


: GF 


25.74 




1.410627 


GB 


34.74 






BF 


9.00 







71 



p. 297.] 



KEY TO ALSOP'S SURVEYING. 



[CHAr. VII. 



Article 406. 



Ex. 2. 



As in Ex. 2 of last b ^ 
article, find AG = 31.01, 
GB = 34.74, GD = 18.76, 
GC = 15.82. 



Hence, 

AsGB 

: GC 

: : GD 

: GH 




34.74 

15.82 
18.76 
8.544 



A. C. 8.459216 
1.199206 
1.273233 



0.931655 



AH = GA - GH = 22.466 
AI = f AH = 12.481, GI = 18.529, and GK = 9.264. 

To find BO, OP, and GL. 



As sin. BOP 


34° 


A. C. 0.252438 


: sin. BPO 


126° 15' 


9.906575 


:: BP 


11.52 


0.061452 


: BO 


16.614 


1.220465 


As sin. BOP 




A. C. 0.252438 


: sin. PBO 


19° 45' 


9.528810 


: : BP 




1.061452 


: PO 


6.961 


0.842700 


As GO 


■ 18.126 


A. C. 8.741698 


: GB 


34.74 ' 


1.540784 


:: GK 


9.264 


0.966799 


: GL 


17.753 


1.249281 



72 



Chap VII.] LAYING OUT AND DIVIDING LAND. 

To find EL and AE. 



[P. 301. 



E]^-hL^ = LG 






1.249281 


Fl^ - LIST = LG - 


-2P0= 3.831 




0.583312 

2)1.832593 




EL = 8.247 




.916296 


AE = AG - (EL 


+ LG) = 5.01. 
Article 407. 






Ex. 2. 








r sin. E 


77° 


A. C. 0.011276 


128° 45' 


u 


" 0.107970 


J sin. A 


76° 




9.986904 


* [sin. D 


78° 15' 




9.990803 


(AD 


8.35 




0.921686 


'' ' [AD 


a 




0.921686 


: Eourthterm 87.162 




1.940325 




174.324 






3BC2 


119.07 

5)293.394 






EE= x/ 58.6788 = 


7.66. 




And 








As sin. G 


25° 45' 


A. 


C. 0.362065 


: sin. E 






9.988724 


: : EE - BC 


L36 




0.133539 


: BE 


3.05 




.484328 


AE = AB - 


BE = 1.60. 






Ex. 3. 








* f sin. E 

Ass . -ci 

t sin. E 


77° 


A. 


C. 0.011276 


78° 15' 


(( 


" 0.009197 


f sin. A 
• t sin. B 


76° 
128° 45' 




9.986904 
9.892030 


r AB 
• • I ^B 


4.65 
4.65 




0.667453 
0.667453 


: zy^ 


17.152 




1.234313 



■r. 302.] KEY TO ALSOP'S SURVEYING. [Chap. VII. 

I /^X2f + ^CJ)\ // 51.456 + 106.58x ^ ^^ 



-- - W V 5 


J~^\ 


5 ^~"-" 


And 






As sin. G- 


24° 45' 


A. C. 0.378139 


: sin.'E 


77° 


9.988724 


: : CD - EF 


1.68 


0.225309 


: FA 


3.91 
Article 408. 


0.592172 


, r sin. E 

Asi • 17, 
t sm. F 


65"^ 15' 


A. C. 0.041846 


90° 


'' " 0.000000 


/ sin. A 
• I sin. B 


76° 
128° 45' 


9.986904 
9.892030 


r AB 

• * I AB 


4.65 
4.65 


0.667453 
0.667453 


: xy^ 


18.017 


1.255686 


, f sin. E 

As i . -c. 
I sm. F 




A. C. 0.041846 




" " 0.000000 


f sin. C 
• t sin. D 


77° 
78° 15' 


9.988724 
9.990803 


f CD 
• = t CD 


7.30 


0.863323 


7.30 


0.863323 


: vw"^ 


55.978 


1.748019 


EF-J(^^^^ 


-1- 2 ?'?r^\ 




^ ^ ^"^ ) = ^ 33.2014 = 5.762. 
5 ^ 


As sin. E 




A. C. 0.041846 


: sin. B 




9.892030 


: : AB 


4.65 


0.667453 


: BO 


3.993 


0.601329 


And 






As sin. G 


24° 45' 


A. C. 0.378139 


: sin. E 




9.958154 


: : EF - BO 


1.769 


0.247728 


: AF 


3.837 


0.584021 



CHAPTER VIII. 



MISCELLANEOUS EXAMPLES. 



Ex. 1. Make AB = 50, one ^^s- 22. 

of tlie given sides; erect the ,'"' ""^-^. 

,. , ^^ 2 Area 
perpendicular BE = 



AB 

19.2. Ty^itli the circle A, and 
radius equal to the other given ~jy 
side, (32,) describe an arc cut- 
ting EC, a parallel through E, in C and C^* then will 
ABC or ABC be the triangle required. 




Calculation. — AD = s/AC^ 



My = v/ 655.36 = 25.6, 

and (12.13.2) BC^ = BA^ + AC^ ± 2 BA . AD = 2500 + 
1024 ± 2560 = 6084, or 964 ; 

whence BC = 78, or 31.05. 



Ex. 2. Let ABCD be the 
o;arden. Produce AB, makins: 
BI = I AB. On AI describe 
the semicircle ALI, cutting 
CB produced in L. Make 
AE = AD, and bisect EB in 
M. With the centre M and 
radius ML describe a semi- 
circle cutting AB in F and K. 
Make BH = BK: then will 
AF or CH be the breadth of 
the walk. 




I K 



/D 



p. 303.] KEY TO ALSOP'S SURVEYING. [Chap. VIII. 

For FH = FB . BK = BP = AB . BI = I AB . BC = 
I ABCD ; also, MF = MK, and EM = MB ; whence FE = 
BK = BH; .•.AF = HC. 



Calculation.— BI = | BC = 87.5 ; BL^ = AB . BI = 10500. 



ML = x/MB2 + ]3L2 ^ ^10600 = 102.95. 
AF = AE + EM - MF = 7.05. 



Ex. 3. Similar triangles are as the squares of their 
homologous sides. 

The area of the triangle, whose sides are 3, 4, and 5, is 6. 

Whence 6 ch. : 24 ch. : : 16 : 64, and n/64 = 8, one side; 
6 and 10 ch. are the others. 



Ex. 4. Construction. — Let ABC be 
the plat. Produce the diameter CA 
to E, making AE = i AD. On EC 
describe a semicircle cutting the 
perpendicular DB in F. DF will be 
the radius of the outer circle. 

For, since circles are as the squares 
of their radii, we have 



ABC : IFG : : DC^ : DFl But CD : DF : : DF : DE; 
whence CD^ : DF^ : : CD : DE : : 4 : 5 ; 

.-. ABC : IFG : : 4 : 5. 

Calculation.—^ : 5 : : AD^ : DP ; : 5625 : 7031.25, and 
DI = 83.85; whence AI = 8.85. 




76 



Chap. VIIL] 



MISCELLANEOUS EXAMPLES. 



[P. 303. 



Ex. 5. Let AB represent the tree, and C and 
D the two stations. Then we have ACE = a 
5° ST', ADE = 2° 29', BCE = 67° 43' 30'', CD 
= 100, and CA = 75.. 




As sin. CED 




3° 8' 


A. C. 1.262332 


: sin. CDE 




2° 29' 




8.636776 


: : CD 




100° 




2.000000 


: CE 




79.27 ' 




1.899108 


tan. BCE 




67° 43' 30" 




10.387619 


BE 




193.52 




2.286727 


BE^ 




37451 




4.573454 


As CE + CA 




154.27 


A. 


C. 7.811719 


: CE-CA 




4.27 




0.630428 


CAE 

: : tan. 


+ 

2 

2 


CEA 

87° 11' 30" 

CEA 

29° 26' 3" 




11.309328 


CAE 
: tan. 


9.751475 


CEA 


57° 45' 27" 




As sin. CEA 




57° 45' 27" 


A. 


C. 0.072733 


: sin. ACE 




5° 37' 




8.990660 


:: CA 




75 




1.875061 


: AE 




8.679 




0.938454 


AE^ 




75.32 




1.876908 


EB2 




37451 







AB = 



v/ 37526.32 = 193.7. 



77 



p. 303.] 



KEY TO ALSOP'S SURVEYING. 



[Chap. VIII. 



Ex. 6. Describe the triangle ADE, 
having the sides ED, AD, and DE 
equal to 6.23, 4.95, and 5.62 respect- 
ivel3\ On DE describe the equilateral e^ 
triangle DBE. Join AB, and on it de- 
scribe the equilateral triangle ABC. 

This will be the one required. 

For, since ABC = EBD, ABE = CBD ; whence, in the 
two triangles ABE and CBD, we have two sides and the 
included angle of one equal to two sides and the included 
angle of the other; consequentl}^ AE = DC. 






Calculation, 






AD 


4.95 






DE 


6.23 




A. C. 9.205512 


AE 


5.62 




" " 9.250264 


2); 


16.80 






8.40 




0.924279 




3.45 




0.537819 
2)19.917874 


cos. 1 AED 


24° 31' 29'' 




9.958937 


i\ED 


49° 2'' 58" 






As EB + EA 


11.85 




A. C. 8.926282 


: EB - EA 


.61 




-1.785330 


EAB + 
: : tan. ^ 


EBA 

35° 28' 

EBA 

2° 6' 


31" 
3" 


9.852871 


EAB - 

: tan. ~ 


8.564483 



Then 

As sin. EBA 
: sin. AEB 
:: EA 

: AB 



33° 22' 28" A.C. 0.259541 

109° 2' 58" 9.975540 

5.62 0.749736 



0.984817 



Chap. YIIL] MISCELLANEOUS EZiAMPLES. [P. 304. 

Finally, 



5 



As rad. A. C. 0.000000 

: sin. 60° 9.93T531 

r AC 0.984817 

•• IaB 0.984817 

: 2 ABC 80.754 1.907165 

ABC = 40.377 cli. = 4 A., E., 5.6 P. 

Fis:. 27. 



Ex. 7. 



As sin. AEC 7° 43' A. C. 0.872007 

: sin. ACE 44° 34' 9.846175 

: : AC 60 1.778151 

: AE 2.496333 



As rad. A. C. 0.000000 

: sin. EAF' 37° 43' 9.786579 

:: AE . 2.496333 

: EF 191.83 2.282912 

_30 

Height 221.83 

And 

As rad. A. C. 0.000000 

: cos. A 9.898201 

: : AE 2.496333 

: AF 248.05 2.394534 



p. 304.] 



KEY TO ALSOP'S SURVEYING. 



[Chap. VII 



Ex. 8. Through A conceive the hori- 
zontal plane AEF to pass, and draw BG 
parallel to EF. Then we have FAE = 
87° 45', FAC = 39° 43', FAD = 52° 13', 
BAE = 2° 17', AEF = 54° 23', GBC = 
33° 4', and GBD = 45° 42'. 



First. To find AE. 
As rad. 
: COS. BAE 2° 17' 

: : AB 157 

: AE 156.88 




A.C. 0.000000 
9.999655 
2.195900 



2.195555 



To find AF, FC, and FD. 



As sin. AFE 


37° 52' 


A. C. 0.211955 


: sin. AEF 


54° 23' 


9.910054 


:: AE 




2.195555 


: AF 


208.54 


2.317564 


tan. CAF 


39° 43' 


9.919448 


Elev. ofhill = FO 


172.5 


2.237012 


As rad. 


- 


A. C. 0.000000 


: tan. FAD 


52° 13' 


10.110579 


:: AF 




2.317564 


: FD 


268 


2.428143 


FC 


.172.5 





Height of tower CD = 95.5 

The calculation might have been made by finding BE 
= 6.25, EF = BG = 255.37, GC = 166.26, and GD 
= 261.69. Whence FC = 172.51, FD = 267.93, and CD 
= 95.43. 



80 



Chap. YIII.] 



MISCELLANEOUS EXAMPLES. 



[P. 304. 



Ex. 9. It is readily seen that AC 
: BC : : tan. DHC : tan. DFC : : cot. 
DFC : cot. DHC. But AC : BC : : sin. 
ABC : sin. BAC. ^^Oience 



D Fig. 29. 





- 


f 


^\ 




A 


] 


As cot. DHC 


32° 


A.C. 


-1.795789 


: cot. DFC 


39° 48' 




10.079267 


: : sin. BAC 


90° 




10.000000 


: sin. ABC 


48° 35' 23'' 




9.875056 


As sin. BCA 


41° 24' 3T" 


A.C 


). 0.179506 


: sin. ABC 






9.875056 


:: J\B 


100 




2.000000 


: AC 






2.054562 


tan. 


39° 48' 




9.920733 




94.47 




1.975295 




10 







CD 



84.47 



Ex. 10. Make BD = 9.43, BDE = 

49° 50', DBE = 72° 41', DA = 7.56, 



Fig. 30. 



and AE = 



BE 
BD 



X 8.42 : also make ABC 



will 



= EBD and BAG = BED : then 
ABC be the triangle required. 

For, since ABC and EBD by con- 
struction are similar, we have EB : BD 
: : AB : BC. But ABC = EBD ; whence 
ABE = CBD. Consequently (6,6) ABE 

6 




81 



p. 304.] 



KEY TO ALSOP'S SCIIVEYING, 



[Chap. YIIl. 



and CBD are similar ; whence BE : EA : : BD : DO, or 

BE BE 

AE = — - . DC. But by construction AE = x 8.42. 

BD -^ BD 

Whence DC = 8.42 ; and, since AD = 7.56 and BD = 9.43, 

the conditions of the problem are fulfilled. 



To find ED. 






As sin. BED 


57° 29' 


A. C. 0.074051 


: sin. DBE 


72° 41' 


9.979855 


:: DB 


9.43 


0.974512 


: DE 


10.6762 


1.028418 



To find AE we have by construction BD : BE : : DC 



AE; or, 

As sin. BED 
: sin. BDE 
:: DC 



cos. 1 ADE 
ADE 
whence ADB 



57° 29' 
49° 50' 

8.42 



22° 48' 29" 

45° 36' 58" 

ADE + EDB 



A.C. 0.074051 

9.883191 
0.925312 



: AE 


7.6305 


0.882554 


To find ADE. 






AE 


7.6305 




AD 


7.56 


A.C. 9.121478 


DE 


10.6762 


8.971582 




2)25.8667 




is 


12.93335 


1.111711 


Js - AE 


5.30285 


0.724510 

2)19.929281 



9.964640 



= 95° 26' 58". 



CHAP.Vni.] MISCELLANEOUS EXAMPLES. 


[P 


To find AB. 








As BD + DA 




16.99 A.C 


. 8.769807 


: BD - DA 




1.87 


0.271842 


BAD 4- 
: : tan. — 


ABD 
ABD 


42° 16' 31" 
5° 42' 51'' 


9.958631 


BAD - 

: tan. — 


9.000280 


As sin. BAD 


47° 59' 22" 


0.128998 


: sin. BDA 


^ 


95° 26' 58" 


9.998032 


:: BD 




9.43 


0.974512 


: BA 




12.63 


1.101542 



[P. 304 



To find BC and AC. 



And 



As 


sin. 


C 


• 


sin. 


BAC 


: ; 


AB 




• 


BC 




As 


sin. 


C 


J 


sin. 


ABC 


\ * 


AB 




: 


AC 





49° 50' 
57° 29' 

13.94 



72° 41' 
15.78 



A. C. 0.116809 
9.925949 
1.101542 



1.144300 



A. C. 0.116809 
9.979855 
1.101542 

1.198206 



Ex. 11. Let AB = 50. Bi- 
sect it in D, and makeDF = 42.5 
the half sum of the two sides. 
At D erect the perpendicular 
DG. Make DE = 2 Area -4- 
AB = 32. Lay off" BG = DF. 
With D as a centre, describe 
the arcs FI and GH, the latter ^ 
cutting EC parallel to AB in H. Join DH, and jDi'oduce 
it to I. Draw ICL perpendicular to AB, cutting EC in C. 
Then will C be the vertex of the triangle. 

It is evident, from the construction, that the fease and 




p. 304.] KEY TO ALSOP'S SURVEYING. [Chap. VIII. 

area agree with the data. It remains to be proved that 
AC + BC = 85. 

Put AD = DB = a, DF = BG = 6, DG = c, DL = d, 
and LC = e: then we have a^ -f c^ = 6^; or 6^ — c^ = a^ 

By similar triangles, DP : DH^ : : IL^ : LC^; or b^ : c^ : : h'' 
— d^ : &\ whence & = — -— = & — - — . 

AC^ = AL^ + LC^ = (a + c^)^ + 6^ = a^ + 2 «c? + 6^2 + c=^ - 

—^h^ + 2ad+ ^^ =b^ + 2ad + — ; 

ad 
whence AC = b -{■ -~ 



Similarly BC = 6 - y .-. AC + BC = 2 6 = 85. 



Calculation. 
DG^ = DF^ - DB^ = 1806.25 - 625 = 1181.25. 
Then, As DH^ : DP : : DE^ : IL^ 

or As 1181.25 : 1806.25 :: 1024 : IL^ = 1565.7989; 



whence DL = v/DP - IL^ = 15.5065, 



and AC = >/ AL^H- LC^ = n/ 1640.77654225 + 1024 
= 51.6215; 

... BC = 85 - AC = 33.3785. 

Ex. 12. Make AB = 20 Fig. 32. 

the given side, and at A 
erect the perpendicular AF ^/ 

= 2ABC-r.AB = 7. Di- ''"" 



vide AB in E, so that AE 
:EB::3:2. Take BG ^ ^^" ' 
a third proportional to AE — EB and EB. With the centre 
G and radius GE describe a circle cutting FC parallel to 
AB in Q and C^ Then will ABC or ABC be the tri- 
angle required. 

84 



CiiAP.VIIL] MISCELLANEOUS EXAMPLES. [P. 304. 

For, since AE - EB : EB : : EB : BG, 

by composition, AE : EBi.-EG : BG. 

Alternately, AE : EG::EB : BG; 

whence, by composition, AG : EG : : EG : BG. 

and (F.6) AC : CB, or A'C : C^B : : AE : EB : : 3 : 2. 

Calculation. 
AE = 12, BE = 8, BG = 16, GE = 24, DC = 7; 



whence GD = GD' = v/GC^- CD^ = ^527 = 22.96. 
Then (12.13.2) BC^ = BG^ + GC^ ± 2 BG . GD = 256 + 

576 ± 734.72 = 97.28, or 1566.72. 
Whence BC = 9.86 or 39.58, 

and AC = i BC = 14.79 or 59.37. 



85 



p. 305.] 



KEY TO ALSOP'S SURA^EYING. 



[Chap. Vlll. 



1—1 



m 
OS 

•< 


O 
1— 1 

o 

I— 1 

1-5 
I— ( 






OO 
CO 

as 
1— 1 










OO 

I— 1 


O 

>o 

CM 
)>; 

Co" 
I— 1 






Si 

.2 

is* 


o 
o 

I— 1 


o 

CO 


3 

o 

Ci 

co' 


o 
I—I 


I— 1 


o 
o 

o 
o 




CO 

o 

CO 

--1 










1—1 




o 


CO 




co' 






^ 












I— 1 


s 


I— 1 




to 
oq 

CO 


1— 1 

CO 






ai 




oo 




OO 

I— J 






1^ 


oo 




CO 








s 


CO 


CM 


i-H 


co' 






» 1 

fcO 

o 
m 

<c 

to 

a 
a 

Xi 

o 


p4 

o 

O 


p4 

o 

!— 1 

xn 


o • 

OO 


p4 

o 

I— 1 


O 




U3 

bO 

a 


o 

CO 

GO 


o 

I- 

m 


p4 

o 

CO 


o 

rH(M 

uo 


o 

1—1 




-2 


r-t 


1 


CO 


^ 


1 





00 

OO 






--hi 


•X 


o 




o 


CO 


OO 


t— 


o 


^ 


2 


o-i 


o 


C-) 


»o 


^ 


CO 


o 


OO 


OO 


o 


l^ 


l^ ' 


C5 


lO 


m 


CO 






CM 


04 
























a 












•ii 
























d 












o 
























OQ 












• l-H 












k 
























'^ 


1— 1 












o 


1:- 


C5 


o 


CO 


«f-l 

O 


"* 


t^ 


OO 


r-) 


1—1 






lO 


OO 


o 


^ 


^ 




t^ 


CO 


o 


r-l 


a 




o 


a) 


C-) 


-* 


*r^ 


1—1 


CO 


1—1 


O 


OO 


y 


uo 


o 


1— 1 


C~) 


rH 


o 


'^f 


I— 1 

d 




I— 1 


1—1 


OO 

53 
> 



^ &; 






V 


V 


O 


o 


I— 1 


CO 


o 


o 


Oi 


-* 


OO 


I— ( 



5ta Cl, 



o 

^' 6 

Ph -t-> 



80 



Chap. VIII.] 

Ex. 14. 



MISCELLANEOUS EXAMPLES. 



[P. 30- 



Bearings. 



S. 26i° E. 



K 60J° E. 



K 281° W. 



Dist. 


N. 


s. 


E. 


w. j 


1.53 




1.37 


.68 


■ 1 

i 


19.37 


9.54 




16.86 


1 


3.27 


2.87 






1.56 






(11.04) 




(15.98) 1 



As diff. lat. 
: dep. 
: : rad. 

: tan. bear. 

As COS. bear. 
: rad. 
: : diff. lat. 

: dist. 



12.41 12.41 17.54 17.54 



11.04 

15.98 

K 55° 22^ E. 
^b"" 22' 



19.43 



A. C. 8.957031 

1.203577 

10.000000 

10.160608 

A. C. 0.245405 

10.000000 

1.042969 

1.288374 





Bearings. 


Dist. 


N. 


s. 


E. 


w. 




S. 55° 22' W. 


8.53 




4.85 




7.02 , 




S. 261° E. 


1.53 




1.37 


.68 




K60i° E. 


8.53 


4.20 




7.42 








(2.02) 






(1.08) 



As diff. lat. 
: dep. 
: : rad. 

: tan. bear. 

As COS. bear. 
: rad. 
: : lat. 

; dist. 



6.22 6.22 8.10 8.10 



2.02 
1.08 

K 28° 8' W. 



A. C. 0.694649 

0.033424 

10.000000 



9.728073 

A. C. 0.054604 

10.000000 

0.305351 



2.29 



.359955 



87 



V. 305.] 



KEY TO ALSOP'S SURVEYING. 



[Chap. VIII. 



Ex. 15. A slight investigation ^ig- 33. 

will show that the distances from 
the observers to the point directly 
under the balloon are as the co- 
tangents of the angles of elevation. 
These cotangents are, a = .73412, 
h = .94345, and c = .75996. Hence 
the following construction : — 

Let A, B, and C be the places of 

the observers. Take AE = AB and 

AF = AC. At the points E and F, 

c b 

with radii equal to - . AE and - . AF, 
a a 

describe arcs cutting in G: join AG 

and make ACD = AGE. Then will 

D be the point under the balloon. 

Since the two arcs will cut in G' as well as G, there will 
be two points (D and D') that will answer the conditions. 

Because AGE = ACD, the triangles ADC and AEG are 
similar ; whence AD : DC : : AE : EG : : a : c; and AG : 
AE : : AC : AD .-. AG : AC (AF) : : AE . (AB) : AD. Since, 
therefore, AFG and ADB have the common angle at A 
and the containing sides proportional, they are similar, and 
consequently AD : DB :: AF : FG : : a : 6. 






Calculation. 




To find FG and EG 






As a 


.73412 


A. C. 10.134230 


: c 


.75996 


-1.880790 


' :: AE 


1000 


3.000000 


: EG 


1035.19 


3.015020 


As a 




A. C. 10.134230 


: b 


.94345 


-1.974719 


: : AF ' 


.1690 


3.227887 


: FG 


2171.88 


3.336836 


88 







Chap. VIIL] 


M 


ISCELLANEOUS EXAMI 


>LES. [P. 


To find AEF an 


id FEa. 




AF 




1690 




AE 




1000 


. A. C. 7.000000 


EF 




1180 

2)3870- 


" " 6.928118 






1935 


3.286681 


i^- 


AF 


245 


2.389166 








2)19.603965 


cos J 


AEF 


50° 39' 57" 


9.801982 


AEF 




101° 19' 54'' 




Fa 




2171.88 ^ 




FE 




1180 


A. C. 6.928118 


EG 




1035.19 

2)4387.07 


" " 6.984980 






2193.535 


3.341145 


is- 


Fa 


21.655 


1.335558 

2)18.589801 


COS. J 


FEa 


78° 37' 37" 


9.294900 


FEG 




157° 15' 14" 





AEC = 360° - (AEF + FEa) = 101° 24' 52" = ADC. 

To find DAC and DCA. 

As AD (a) - A. C. 10.134230 

: DC (c) -1.880790 

: : rad. 10.000000 



: tan. 


X 


45° 59' 26" 




10.015020 


As rad. 






A.C 


. 0.000000 


: tan. 


X - 45° 


59' 26" 




8.237799 


: : tan. 


DAC + ACD 

2 
DAC - ACD 

2 


39° 17' 34" 

48' 38" 




9.912902 


: tan. 


8.150701 


DAC 


40° 6' 12" 





89 



p. 305.] 



KEY TO ALSOP'S SURVEYING. 



[Chap. VIII. 



101^ 24^52'' A.C. 0.008677 
40° 6' 12'' 9.808999 

1690 3.227887 

1110.6 3.045563 

52° 46' 10.119210 

1461.4 3.164773 

or AEG' = FEG - AEF = 55° 55' 20" = AD'C. 
Then, as AD' : D'C : : rad. : tan. x. = 45° 59' 26" as before. 

And 

As rad. A. C. 0.000000 



As 


sin. 


ADC 


J 


sin. 


DAC 


• • 


AC. 




: 


CD 




tan 


. 




He 


ight 





: tan. x — 45° 


59' 26" 


8.237799 


D'AC 

: : tan. 


+ ACD' 

2 
-ACD' 

2 


62° 2' 10" 
1° 52' 0" 


10.275037 


D'AC 
: tan. 


8.512836 


D'AC 


63° 54' 20" 




As sin. AD'C 




5."^° 55' 20" 


A.C. 0.081825 


: sin. D'AC 




63° 54' 20" 


9.953311 


:: AC 




1690 


3.227887 


: CD 




1832 


3.263023 


tan. 




52° 46' 


10.119210 


Height 




2411 


3.382233 


Ex. 16. 




' 




To find the area 


of the whole tract. 





Sta. 
1 

2 


N. 


s. 


E. 


w. 


E.D.D. 


W.D.D. 


Multipliers. 


S. Areas. 


10.82 


1 


19.78 




57.71 






15.81 




2.79 




22.57 


22.57 W. 


356.8317 


3 


G.99 




12.48 




9.69 




12.88 W. 


90.0312 ! 


4 
5 

6 

7 




6.75 


15.64 




28.12 




15.24 E. 


102.8700 




1.56 


22.35 




37.99 




53.23 E. 


83.0388 




16.26 


10.03 




32.38 




85.61 E. 


1392.0186 




9.05 




37.93 




27.90 


57.71 E. 


522.2755 ; 

1 



33.62 33.62 60.50 60.50 



90 



2 ) 2547.0 058 
1273.5329 



Chap. VIII.] MISCELLANEOUS EXAMPLES. [P. 305. 

To find the area to a line north, from the first station. 



1 

sta. 
1 

2 
3 
4 


N. 


s. 


E. 


w. 


E.D.D. 


W.D.D. 

19.78 


Multipliers. 


S. Areas. 


10.82 






19.78 




■ 1 


15.81 






2.79 




22.57 


22.57 W. 


356.8317 


6.99 




12.48 




9.69 




12.88 W. 


90.0312 




4.35 


(10.09) 




22.57 




9.69 E. 


42.1515 


5 




(29.27) 






10.09 




19.78 E. 


578.9606 



33.62 33.62 22.57 22.57 



1067.9750 



As rad. 

: cot. bear. 4th side S. 66° 40' E. 

:: Dep. 10.09 

: Lat. ■ 4.35 S. 



A.C. 0.000000 
9.634838 

1.003891 

0.638729 



1273.5329 - 1067.9750 = 205.5579 = double area between 
the division line and a line due north 29.27 ch. from the 
first station. 



Then 






As 1 ^!°- 
t sm. 


76° 35' 


A.C. 0.012017 


66"^ 40' 


" " 0.037055 


( rad. 
\ sin. 


36° 45' 


10.000000 

9.776937 


: : 2 area 


205.5579 


2.312934 


: fourth term 


137.703 


2.138943 


(29.27)^ = 


856.7329 






v/719.0299 = 


26.815 = division li 


As sin. 


36° 45' 


A.C. 0.223063 


: sin. 


GQ"" 40' 


9.962945 


1 1 


2.455 


0.390051 


: Dist. from 1st cor. 3.77 


0.576059 



Ex. 17. As it is of no importance to the calculation of 
this question what the course of the line is, we shall sup- 

91 



p. 306.] 



KEY TO ALSOP'S SURVEYING. 



[Chap. VIII. 



pose it to be north. The bearings of the difierent lines in 
course will then be as below. 



Sta. 


Beariugs. 


Dist. 


N. 

3.81 


s. 


E. 


w. 


1 

2 
3 
4 

5 


South. 




(23.67) 


9.24 




K 67° 35' E. 


10.00 






K 18° 52' E. 


7.25 


6.86 




2.34 




K 7° 7'E. 


5.43 


5.39 




.67 




K 58° 10' W. 


(14.42) 


(7.61) 






(12.25) 



As rad. 






A.C. 


0.000000 


: cot. bear. 5th side 


58° 10' 




9.792974 


: : Dep. 




12.25 




1.088136 


: Lat. 




T.61 




0.881110 


As sin. bear. 






A.C 


. 0.070793 


: rad. - 


\ 






10.000000 


: : Dep. 








1.088136 


: Dist. 5th side 




14.42 




1.158929 


Distance of line, 


23.67 


. 







Fig. 34. 



Ex. 18. Here we have AE : CE : : 
cot. CAE : rad., and DE : EC : : cot. 
ODE : rad. .-. AE : ED : : cot. CAE 
: cot. CDE. Whence cot. CDE : cot. 
CAE : : sin. DAE : sin.' ADE. 




As cot. CDE 


20° 4' 


A.C. 9.562636 


: cot. CAE 


25° 10' 


10.328037 


: : sin. DAE 


90° 


10.000000 


: sin. ADE 


51° 1'40" 


9.890673 


92 







Chap. YIIL] MISCELLANEOUS EXAMPLES. [P. 306. 

As rad. A. C. 0.000000 

: tan. ABE 10.092061 

: : AB 60 1.778151 



: AE 


74.17 


1.870212 


As rad. 




A. C. 0.000000 


tan. CAE 


25° 10' 


9.671963 


: : AE 


- 


1.870212 


: EC 




1.542175 


As EC 




A. C. 8.457825 


: EB 


4.17 


0.620136 


: ; rad. 




10.000000 


: cot. EBC 


83° 10' 34'' 


9.077961 


Ascos. EBC 




A. C. 0.925118 


: rad. 




10.000000 


: : BE 




0.620136 



BC 35.1 yds. 1.545254 



93 



p. 306.] 



KEY TO ALSOP'S SURVEYING. 



[Chap. VIII. 



Ci 



X 
^ 



o 

CO 



I I 



<M 



O 

o 

CO 






^' 


S 


^ 


W' 


f4 


t- 
t^ 


rH 


o 


^1 

00 


oo 
I— < 


t^ 

^H 


t- 


o 

CO 


G<1 

CO 


CM 



rH 






o 



24.98 


^. 

CO 

CM 


O 
CM 

c4 





o 



o 

c-j 



CO 

o 





rH 


l~- 


-Tf^ 






w 




I— 1 


oq 














CO 


r^ 






02 






rH 
I— 1 


o 
o 

rH 


1-4 





1 G<1 



Ci OC 

CO l^ 

lO <M 



&q 


!^" 


s 


^' 


1 


V 




V, 


V 




O 




urs 


r^. 


C? 


-^ 




I— 1 


CN 


-u> 


o 


o 


o 


o 


• o 




O 


CO 


C7i 


CJ 


t-~ 


CO 


rH 


^ 


t^; 


CJQ 


m 





^ 


^ 


H* 


p4 


W 


V 




V 


V 




O 




lO 


O 




(M 




-* 


^ 







o 


o 


o 


o 


.-^ 


O 


O 


O 


CO 


O 


rH 


CO 


O 


OO 


'^. 


^ 


!z; 


'n 


02 



1 


c.t 


-Tf 


UO 



rH O 

lO CM 

O CO 

1-. o 

ci CO 

CM t— 

(M CM 



OO 

I- 

o OO 



'^ CM 



00 
OO Oi 

o t- 



O CO CM 

CO* O 



<M 





o" 


CM 


o 


r— 1 


CO 




o 


o 




CO 






o 


o 


(X) 


OO 




. 


o 


o 




-H 




M 


<>J 


o 


r-T 


-TtH 




^ 


t- 


o 


t^ 


-^ 






o 


o 
1—1 


d 


J—{ 




OC 


Q 










c3 
a> 


< 










u 












c3 












© 
























rQ 












g 












o 


T-A 






CO 




ft 


o 






o 
b^ 

<M 


CM 












OO 




t^ 








CM 




d 








CO 




© 








(M 




02 


Tl 


-119 


-4^ 


0(3 
(M 




O 


CJ 


r^ 


'TU 


CO 




<1 


•• 


.. 


•• 


'^fl 












CO 












05 












CM 












-t^ 




cri 


05 


f-1 


00 


CO 




OO 


-* 


<— > 


CO 






I— I 


CM 


c— ) 


■^ 


CM 




CO 


-^ 


(-) 


CM 




t^ 


CO 


(— ) 


rH 






CM 


'^ 


O 


t^ 






a> 


i-H 


o 

T-H 


o 

T-H 






O 












-< 









t^ OO 

CM '-J 

CM 



iO 



^ W 



V 




o 




.— 1 




r— 1 




o 


o 


O 





Oi 


fO 


>o 


<_) 


r- 


on 


o 


OC 


, 




I— 1 


rH 


;2; 


CO 







OQ 

d 

rd 

o 

CO 

(M 

V 

Oi 

U3 



125 

OQ 

o 

p 

^^ 

d 

.2 

lO ^ 
O -^ 



02 



© 



.S 

c3 



© 

© 

g 



94 



CHA.P. VIIL] 



MISCELLANEOUS EXAMPLES. 



rp. 306. 




Ex. 20. Let ABCJ) be Fig. 35. 

the quadrilateral. Divide ^ 

it into two equal parts by 

the line EF parallel to CD. ^- \Q 

(Art. 407.) Then divide 

the quadrilaterals ABCD ^^ ,^ ^j^ 

and AEEF into two equal ----- \n 

parts by the lines GH and g:.^— :.____^^_^_-_._ _^ii 

IK parallel to AD. Join "^^s^^-z^"':^ 

HK, and produce it to M. ' '' \ p ' 

Through G draw GL pa- a— — -^ ^ 

rallel to MH. Then will 
ML be the other division line. 

For, since GL is parallel to MH, the triangles MGH and 
MLH are equal : to each add AMHD, and we have AMLD 
= AGHD = lABCD. 

Again, since XK is parallel to LH and FN" to GL, we 
have MX : XL : : IsIK : KH : : MI : IG. 

Whence IX is parallel to GL. The triangle MIK is 
therefore equal to MXK : to each add AMKF, and we have 
AMXF = AFKF = 1 ABCD. 



To find AF and EF. 




* f sin. C ' 
As<^ . -p. 
( sm. D 


78° 45' 


A. C. 0.008426 


63° 


" " 0.050119 


' sin. A 


78° 30' 


9.991193 


( sin. B 


139° 45' 


9.810316 


f AB 


23 


1.361728 


'•'' \AB 




1.361728 


: fourth terra 


383.274 


2.583510 


CD^ 


2161.3201 
2)2544.5941 




EF 


x/ 1272.2970 = 


35.67. 



95 



306.] KEY 


TO ALSOP'S SURVEYING. 


[Chap. VTII 


As sin. 


38° 15' 


A.C. 


0.208243 


: sin. C 


78° 45' 




9.991574 


: : CD. - EF 


10.82 




1.034227 


: FD 


17.14 




1.234044 


AD 


49.64 






AF 


32.50 






To find AGr and GH. 






fsin. A 
^' I sin. D 




A.C. 


0.008807 




a (( 


0.050119 


r sin. B 
* I sin. C 






9.810316 
9.991574 


/BC 
'' IBC 


30.50 . 




1.484300 
1.484300 


: fourth term 


675.15 




2.829416 


AD^ 


2464.1296 






2^ 


(3139.2796 






GH = ^/1569.6398 = 39.62. 




As sin. 


38° 30' 


A.C. 


, 0.205850 


: sin. D 






9.949881 


: : AD - GH 


10.02 




1.000868 


: AG 


14.34 




1.156599 


Draw EQ parallel 


to AD. Then 






As sin. CEQ 


38° 15' 


A.C. 


, 0.208243 


: sin. CQE 






9.949881 


:: CQ 


"10.82 




1.034227 


: EC 


15.57 




1.192351 


BC 


30.50 






BE 


14.93 







96 



Chap. VIII.] MISC 


ELLANEC 


To find AI and IK 


, 


As / ''''• ^ 
' ^^ \ sin. E 




r sin. B 
• tsin. E 




(BE 
'' IBE 


14.93 


: fourth term 


161.79 


AE^ = 


1056.25 



[P. 306. 



A. C. 0.008807 
" " 0.050119 
9.810316 
9.991574 
1.174060 
1.174060 
2.208936 



2)1218.04 



IK = v^609.02 = 24.68. 



As sin. 


38° 3 


: sin/E 




:: AE - IK 


7.82 


: AI 


11.19 



A. C. 0.205850 
9.949881 
0.893207 
1.048938 



^Now, because IK is parallel to GH, we have 
GH : IK : : MG : MI; whence, by division, 
aH-IK:IK::ia:MI; or 
14.94 : 24.68 : : 3.15 : MI = 5.20 ; 
and AM = AI - IM = 5.99. 

Through M and H draw MO and HP parallel to AD and 
AB. Then in HPO we have 

As sin. HOP (D) A. C. 0.050119 

: sin. HPO (A) 9.991193 

: : HP 8.35 0.921686 

: HO 9.18 0.962998 



As sin. HOP A. C. 0.050119 

: sin. PHO 38° 30' 9.794150 

: : HP 0.921686 

: HO 5.83 0.765955 
And by similar triangles MOH and GHL we have MO 

(45.45) : OH (9.18) : : GH (39.62) : IIL = 8. 

7 97 



p. 306.] 



KEY TO ALSOP'S SURVEYING. 



[Chap. VIII. 



Then in MOL we have MO, OL, and MOL, to find 
OML. Thus 

8.203218 
1.451326 

10.212681 
9.867225 



As MO + OL 




62.63 


: MO - OL 




28.2T 


MLO + OML 
::tan. ^ 


58° 30' 


MLO - 


OML 


36° 23' 


. tan. ^ 
OML 




22° 7' 


Bearing of MO 


S. 


66^ 15' E. 


" " ML 


S. 


88° 22' E. 



98 



THE END. 



BTETJSOTYPED BY L. JOHNSOX & CO. 
PUILADEU?HIA. 



m'^' 6 A 8 



0) 



Deacidified using the Bookkeeper process. 
Neutralizing agent: Magnesium Oxide 
Treatment Date; Jan. 2004 L 

PreservationTechnologJes * 

A WORLD LEADER IN PAPER PRESERVATION j 

1 1 1 Thomson Park Dnve . 

Cranberry Townshin. pa isnfiR i 



LIBRARY OF CONGRESS 










